T H E C O H O M O L O G Y OF KONTSEVICH'S S T A C K O F S T A B L E M A P S T O P*, T H E C A S E OF CONICS. by ANNE FIONNUALA O'HALLORAN B.A. (Mathematics and Legal Science) The National University of Ireland (Galway), 1992 M.A. (Mathematics) The National University of Ireland (Galway), 1993 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R O F PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Mathematics We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A November 2000 © Anne Fionnuala O'Halloran, 2000 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Date Abstract In this thesis we consider the singular cohomology of M , o ( P , 2), the coarse moduli space associated to Kontsevich's stack of degree two stable maps to P , Mo,o{P , 2). We show that the cohomology ring is generated by a divisor d which corresponds to the locus of pairs (C,g) with C reducible, and the first and second Chern classes, C\ and C2, of the canonical rank three vector bundle E = 7r*/*0(l) on A^o,o(P , 2), where 7r is the canonical projection associated to the universal curve C and / is the universal map. We give the cohomology ring as a quotient of a polynomial ring in these generators. The relations are in degrees n, n + 1 and n + 2. n 0 n n n We also give a representation of the cohomology ring in terms of the Chern roots of E. The results are conjectural for n » 0. s ii Table of Contents Abstract ii Table of Contents iii Acknowledgement v Chapter 1. Introduction 1 Chapter 2. Coordinate charts for M 2.1 2.2 ( P , 2) 14 n 0 ) 0 What is AT o(P ,2)? G and A actions on P 2.2.1 G and A actions 2.2.2 Two special actions on P™ 2.2.3 A BB-minus decomposition of P 15 20 20 21 26 n 0) 1 n m 1 m 2.3 2.4 2.5 2.6 2.7 G and A actions on_A* o(P > 2) A n open substack of_Alo,o(P ,2) An Open Cover for -M ,o'(P , 2) Covering the Fixed Locus 2.6.1 Transition Functions 2.6.2 The Fixed Locus A vector field on M>,o(P , 2) 1 3.2 0l n n 0 n The Cohomology of M ( P , 2) The Hypercohomology of AT ,o(P ,2) 3.1.1 Preliminary Ideas 3.1.2 The Hypercohomology Calculation 3.1.3 The Case n =_2_ The Cohomology of M ( P , 2) 91 91 100 119 128 0 n 0 ) 0 Geometric Interpretation 132 Chern Classes Chern Classes and Hypercohomology Some Divisors on A^o.o(P ,2) ^ Chern Classes and the Cohomology of M , o ( P , 2) Geometric Interpretation in the case n = 2 4.5.1 M ,o(P ,2) as the Blow-Up of the Veronese Surface in P 4.5.2 Cohomology of M , ( P , 2 ) via the Veronese 4.5.3 Geometric Interpretation for n — 2 Chern Roots and the cohomology of M ,o(P", 2) 71 n 0 2 0 2 0 4.6 91 n 0 ) 0 n Chapter 4. 4.1 4.2 4.3 4.4 4.5 28 31 72 74 76 81 82 n m Chapter 3. 3.1 n 0 0 iii 5 133 136 148 150 151 151 154 158 159 Table of Contents Bibliography Acknowledgement I would like to thank my research supervisor, Dr. K . Behrend, for all his help and support. I also thank Dr. J. Carrell and Dr. W. Casselman for their careful reading of this thesis. Thanks to my friends and family for their patience, particularly Russell Joyce, Sonia Pawlak, Heather Ferguson, Christine Ferguson, David Burggraf and K . G . O'Halloran. Chapter 1 Introduction Suppose we consider conies in projective n space, P . If n = 2 a conic is given by a degree n two homogeneous equation, if n > 2 a conic is given by a collection of homogeneous polynomials. Instead of focusing on the polynomial or polynomials giving the conic, one can represent the conic using a parameterization. Specifically, if we consider a fixed nondegenerate conic in P n (non-degenerate meaning that it is not given geometrically by a line or a pair of lines), one can find a degree two morphism / from P to P 1 such that n the image of P under / is this fixed conic. So / is of the form 1 f :< s,t >—K aos + bost + cot , ais + bist + 2 2 2 Cii ,... , 2 as 2 n + b st + c t 2 n n > where the coordinate polynomials do not simultaneously vanish. Thus a non-degenerate conic in P n is given by a pair (C, /) where C = P 1 and / is a degree two morphism from P to P . Suppose we also consider pairs (C, /) where f(C) is a line in P and / is 1 n n as above, then / gives a two to one branched covering of this line with two branch points. We consider the coarse moduli space, M o ( P , 2), which parameterizes such pairs (C, / ) . n 0i The space M ,o(P , 2) has a natural compactification M , o ( P , 2), due to Kontsevich [12], n n 0 0 where the points of M o ( P , 2) also include pairs (C, /) where C is two copies of P meetn 1 0) ing transversally and / : C -> P is a morphism such that if we restrict / to either copy n of P 1 the restriction is an isomorphism. The space M , o ( P , 2 ) is actually the coarse n 0 moduli space associated to the stack of stable maps A^o,o(P ,2). For the properties of n 1 Chapter 1. Introduction Mo,o(P",2) and ^f o(P ,2) see [8] or [3]. n 0> Suppose we consider pairs (C, /) where C is isomorphic to P and / is a degree d mor1 phism from C to P , i.e. / is of the form n f :<s,t >->< / (s, t), fi(s, *),..., f (s, t) > 0 n where the /j are degree d homogeneous polynomials which do not simultaneously vanish. The coarse moduli space M ,o(P ,rf) parameterizes such pairs. Again we have a natural n 0 compactification M ,o(P ,d) n 0 due to Kontsevich [12]. Kontsevich's stacks of stable maps are central to the ideas of quantum cohomology. The coefficients in the multiplication table of the quantum cohomology ring are the solutions to certain enumerative geometry problems. This is in general quite remarkable. Of course quantum cohomology is guided by ideas in physics, and in particular has applications in string theory. Therefore understanding the structure of these stacks of stable maps has become very important. In this thesis we calculate Hg (M o(P , 2)(C), C), the singular cohomology ring of n ing 0> Mo,o(P , 2). The method we use is one which we shall attempt to use in the future to n calculate the singular cohomology of M ,o(P ,d), n 0 for d > 2. We try to make the theory we develop in this thesis as general as possible, since the case of conies is really a test case. There is almost no information on this cohomology ring in the literature. numbers of Hg (M fl(P , n ing Q The Betti 2)(C),C) were calculated previously in [9] by Getzler and Pandharipande, but their methods give no information about the ring structure (although using the Betti numbers one might make some conjectures about the number of generators and relations). In [14] Pandharipande computes the Picard group, the Picard 2 Chapter 1. Introduction group has two generators, the first is a divisor corresponding to the pairs (C, /) with C reducible, and the second is a divisor corresponding to pairs (C, /) where f(C) meets a fixed codimension two linear subspace of P . Again we cannot deduce any information n on the ring structure. The coarse moduli space M o(P", 2) associated to the stack of stable maps Mo,o(P , n 0) 2) has a classical analogue, namely the space of complete conies in P . This classical space n specifies a plane, together with a complete conic contained in the plane. For n = 2, the plane is always just P , so in particular the space of complete conies in P is the same 2 2 as M ,o(P ,2). For n > 2, Kontsevich's space has blown down all the planes containing 2 0 a given line. For d > 2, the coarse moduli spaces M ,o(P ,d) do not have a classical n 0 analogue. The scheme M ,o(P ,2) has a second well known representation. Suppose we consider 2 0 the Veronese surface in P , then Mo,o(P ,2) is isomorphic to the blow-up of P along 5 2 5 the Veronese surface, P . Although this is a well known fact, we could not find a proof 5 in the literature. For this reason we provide an explicit isomorphism in Lemma 15, Section 4.5.1. One can easily calculate the Chow ring of P , and thus the Chow ring of 5 M o ( P , 2 ) . For M ,o(P ,2) the Chow ring and the cohomology ring coincide, so we can 2 0) 2 0 calculate the cohomology ring of M o ( P , 2 ) . We do this in Section 4.5.2, and verify that 2 0) we get the same result as when we calculate using our method described below. We now state the main result of this thesis. Let n > 2. We have a canonical rank three vector bundle E = T T * / * 0(1), on A^ ,o(P ,2), where 7T is the canonical projection n 0 associated to the universal curve C for Mo,o(P , n for E over a point a is H°(C , a 2) and / is the universal map. The fiber f*0(l)), where C is the curve corresponding to a. Let a 7i, 72 and 73 be the Chern roots of E, and let the group with two elements, Z , act on 2 3 Chapter 1. Introduction C[7i, 72,73] as follows: 71 -> 7i 72 -> 73 73 ->• 72- Then #WMo,o(P ,2)(C),C) = (C[ , , ]/(R ,R ,R )f n 7l l2 l3 1 2 3 where R i = 7" ~ 7? +1 2 , 7" +1 72 - 7 i «, = +(7 Rs = 72-Ti ( 7 _ 5 ] ) _(fc^) \ +1 2 7 / + 7 r ) + (7 n+1 +1 73 ~ 7i _(fe^) V ~ 7i" +1 3 ( 73-7i 7 _ 2 7 l ) / + 7i ) n+1 +1 3 7 r ( ( 7 2 - 7 i ) + (73-7i)). +1 We also give the cohomology ring in two other forms. The first involves the Chern classes, the second is not motivated by geometry but is the simplest representation. Let ci and C2 denote the first and second Chern classes of E and let d be the divisor of points (C, /) 6 M , o ( P , 2) such that C is reducible, where c\ and d have degree n 0 one and C2 has degree two. Then the cohomology ring #* (M o(P , 2)(C), C) is n infl 0j C[d, ci, c ]/{g i, fn+i, r +i) 2 n+ n where ^ 9n+l x ( -d +4c 2 fn+l 1 2 d-2g 6 3 , 2 2c 2 V d+4ci . d +16c?+8dci 2 -4c H 6 2 0 4 j Chapter 1. Introduction Finally, if we let d- 2ci u 3 d + ci b d 2 -4c 9 2 1 3 4c? 1 3 then H*(M ,o(P ,2)(C),C) equals n 0 C[6,«, # ] / ( n + l , / n + i , r „ + i ) s where / fn+1 \ n+l J r = 0 |« -2 2 is \ 0 n-l 1 \ / « + 26 5 + 46 \ 2 \ 6 («-26) y 0 2 In terms of the Chern roots, 7i 72 2 72 These representations are conjectural for n » 0. The method we use to calculate the cohomology ring of M ,o(P ,2) is similar to one n 0 which has been used in the past to calculate the cohomology of a smooth complex projective variety X, in the situation where we have a vector field on X with isolated non trivial zero set Z (we describe this method below). In our case X is an stack (which complicates matters) and we have a vector field on X with a one dimensional zero locus. As far as we know, this is the first time a method of this type as been used in the case 5 Chapter 1. Introduction where the vector field has a strictly positive dimensional zero locus (and the calculation is non-trivial). We now describe our method. Let X be a smooth and proper Deligne-Mumford stack, and X its associated coarse moduli space. The singular cohomology Hg (X(C), C) of X(C) with its C-topology is ing equal to the derived functor cohomology of the analytic space X(C) with the constant sheaf C, H*(X(C), C). This in turn equals the sheaf cohomology of the analytic stack Xanai, H*(X i,C) ana [2]. We consider the holomorphic deRham complex c -* o -+ si Xhol Xhol -+ tf Xkol -» • • • which is exact. Thus the map of complexes 0 -+ -> n °x xhol hol t 0 -+ t c -»• t 0 -)• 0 is a quasi isomorphism, and the induced map on hypercohomology H* (x , q -> ET (x ,, o ana anal Xhoi -+ n ->•••) = Xkol (x , 04 j anai is an isomorphism. So in particular H* (x(c), c) = #*(x ,, q =ff(x , c) ana sing IT (x ,, n^), anai ana is an isomorphism of rings. We have a canonical spectral sequence 'E*!' — H (X i,DF ) q 9 ana to the hypercohomology W (X i, +q ana £l* ). for p, q > 0, which abuts Since X is a smooth and proper Deligne- hol Mumford stack, G A G A applies [15, 2], thus H<(X ff )XH<{X W) mall hol hal 1 6 Chapter 1. Introduction where the latter sheaf is the p th exterior power of the algebraic cotangent sheaf. In the case where we have the added assumption that H (X,Q ) q p = 0 for p^q (this is true for the stack A^o,o(P , 2), see Lemma 36), the spectral sequence 'E^ degenn q erates at E\ and M*(x ,n* ) = anal ® H (x,n ). k hol k k Thus we have as rings. The idea at this stage is to choose a different complex of sheaves KP for p < 0, on X, with an O^-linear differential, such that we have a spectral sequence E abutting to r the hypercohomology H* (X, fC*) with the property that for q > 0 and p < 0, { H (X,tt- ) forp + q = 0, 0 otherwise, g p and whose hypercohomology is accessible. In this case H*(X,/C*) = 0 We explicitly calculate If(X, filtration F of If{X, 9 for k^O. JC*). By standard spectral sequence arguments we have a /C*), such that F^(W(X,JC*)) ~ H 1 Therefore the associated graded ring Gr (lf(X,K:*)) F ' ' j is isomorphic to @ H (X, Thus Gr (W(X,lC*)) ^ F 7 H; (X(C),C). ing g q Q ). q Chapter 1. Introduction We do not have to compute this associated graded ring because if we choose our complex of sheaves /C* correctly, (using a result of Carrell and Akyildiz [1]) it turns out that W(X,K*) is already graded (from a G action) and the filtration F from the spectral Q m sequence equals the filtration from this grading. Therefore H pf./C*) = Gr (H° ( * , £ • ) ) = 0 F H* (X(C),C) sing as graded rings. The complex we choose is the global Koszul complex associated to a vector field V. We first describe this notion for X a smooth complex projective variety and V a vector field with isolated but non trivial zero set Z. What we describe in this setting in due to Akyildiz and Carrell [1]. Given an A 1 G action on X we have an associated vector field V. action on X such that these actions constitute an A , G 1 m m a positive integer k such that for A e G , m dX.V = X V. k Suppose we have a pair, i.e. there exists We consider the contraction operator associated to V, which has the property that i(V) 2 = 0. Thus we have a complex of sheaves on X, K? = Q,~ for p < 0, with differential i(V). p Let Ui be an affine open cover for X, such that Q, restricted to Ui is trivial. The vector field on Ui is given by a collection of regular functions, (v\,... ,v ), n for Ui where n — d i m X The complex IC restricted to Ui is just the Koszul complex of the ring 17(17,) with respect to the sequence (vi,... ,v ). n If V has an isolated zero in Ui then, since 8 Chapter 1. Introduction r([/j) is Cohen-Macauley, (vi,... , v ) is a regular sequence, so the complex restricted to n Ui is exact except at position 0, and the homology at position zero is T(U)/(vi,... ,v ). n If V does not vanish on U{ then the complex is automatically exact. Thus we have a quasi isomorphism of complexes ••• -> fix -> x I I 0 and the sheaf Ox/i(V)Q, x IP(X, (...-• 0 0 O -> o /i(v)n x -»• o x is zero away from the fixed locus. Now Oxli{V)Sl )) = H°(X, O /i(V)Q ) x x x = H°(Z, O ), z where Oz is the structure sheaf of the fixed locus viewed as a (possibly non-reduced) scheme, i.e. Oz = Ox/i(V)Q , so x H°(X,/C*) = We have a spectral sequence 'E{ ,q z = H (X, Q, ) abutting to the hypercohomology q p = 0 for p + q ^ 0, this spectral sequence degenerates at 'E\, so W (X, fC). Since 'E{ +q H°(Z,O ). ,q in particular M°(X,/C*) = H°(Z,O ) has a filtration F such that q z F (H°(Z,O )) q z F^{H\Z,O ))~ ' [ z ' J Carrell and Akilydiz show that this mysterious filtration is just the canonical filtration given where F q is the q th weight space of the induced G m H°(Z, O ) is already graded and H* (X(C), z ing action on H°(Z,Oz)- Thus C) is isomorphic to H°(Z, O ) as a graded z ring with the grading induced on the latter by the G m action, i.e. H; (X(C),C)=H°(Z,O ) ing z as graded rings. In fact Carrell and Akilydiz show that even in the case where the dimension of Z is strictly positive we have such a result. In this case we have H; (x(c),c)=®P(x,)c*) ing Chapter 1. Introduction as graded rings. In our case X is the stack of stable maps Mofi(P ,2). In order to work with this n stack we construct for it an etale open cover. Specifically, we find a collection of afl&ne varieties, {Ui}, each isomorphic to A ~ Zn and each equipped with a morphism of stacks l (j>i:Ui^M ,o(P ,2) n 0 such that <f>i is etale, and the collection taken together map surjectively to Mo$(P , n Each Ui is equipped with an action of Z 2 2). (the group with two elements) such that the induced map from the quotient stack, [Ui/Z ] 2 is an open immersion, cf>:[Ui/Z }^Mo,oCP ,2). n 2 It is well known that such covers exist for these stacks, however one has not been explicitly constructed in the literature before now. Chapter 2 of this thesis focuses on the construction of this etale open cover. The next task is to find a vector field on A (o,o(P , 2) and thus on the cover {Ui}. We / n first consider a vector field on P", which in turn induces a vector field on M-o,o(P , 2). n Suppose we let A 1 act on P n via the matrix ( n! t~ t~ n 2 (n-2)! M(t) = \ \ (n-l)! n x (n-l)! 0 0 1 t m-3 LTI-2 (n-3)! (n-2)! 0 0 0 0 1 0 0 0 0 0 10 Chapter 1. Introduction then by differentiating we have a vector field V oh P , which has a unique zero. We have n an induced A 1 (C,f) action on the stack A io,o(P \2) which is described on geometric points / 7 ofM),o(P ,2)(C) by n t.(C,f) = (C,M(t)of). The fixed locus consists of closed points (C, /) such that f(C) is a conic in P n which is invariant under the action of A , such conies are paramaterized by A , plus one other 1 1 closed point (C, f) where C is degenerate, / ( C ) is the unique A 1 invariant line in P , n and / maps the crossing point to the unique A fixed point in P . 1 n We have an induced vector field on Afo,o(P , 2), and an associated global Koszul complex n with differential i(V) as before. Since the maps fa : Ui -> A1o,o(P > 2) are etale, we have n in particular and for the purposes of calculating hypercohomology we need only look at the Koszul complex associated to the induced vector field on the Ui (in each case this induced vector field is given by a collection of regular functions on Ui), and at the gluing data for these Ui. For each Ui which does not meet the zero locus of V, the Koszul complex on Ui is exact as in the scheme case. We need two neighborhoods to cover the fixed locus, we call them UQ and U\. Suppose the vector field is given by (v^i,... ,i>i,3 -i) n on Ui. For i = 0,1, we can find a subsequence of regular functions of length 3n — 2 such that this subsequence is a regular sequence (it is this part of the proof which remains conjectural for n » 0, since verifying that we have a regular subsequence is computational). So the Koszul complex is exact except in positions zero and one in each case. Thus we have a 11 Chapter 1. Introduction quasi isomorphism of complexes of sheaves % ,o(P ,2) I 0 0 I ^o,o(P«,2)/(^(^)%, 0 ~* n n 0 I ••' °M ,o(P ^) ^7CT ,o(P ,2) ~* n 0 o ( p n , 2 ) ) ^ o o ( p n ) 2 ) -> 0 and to calculate the hypercohomology H°(7Vt ,o(P ,2),/C*) we need only calculate the n 0 hypercohomology of the lower complex restricted to UQ and U\. Thus we are reduced to considering the following four term double complex: ... -+ 0 0 0 t t (^u /(i(V)n ))(U ) -> 2 0 UO 0 (U ) -+ 01 UO t •- 0 : ©< = 0 l l -> OL 0 (*) t (^/(*(^)^))(^) ^ ©<=o,iO^(^) -> 0 Even when the double complex is reduced to four terms, calculating the hypercohomology is a difficult task. We introduce the idea of a companion pair of vector fields for V in Chapter 3 (Definition 11) as a simplification technique. One can define an action of the group Z x Z on each of the four terms of the double com2 2 plex (*) such that with respect to this action all the maps are equivariant. Thus taking the Z x Z invariant part of the degree zero hypercohomology, denoted by H ° ( * ) 2 Z 2 x Z 2 2 , is the same as taking the degree zero hypercohomology of the invariant subcomplex. This ring H ° ( * ) Z 2 x Z 2 maps injectively to the degree zero hypercohomology of the double com- plex (*), thus maps injectively to EP(A7 ,o(P , 2), JC*). n 0 We next find a ring R(n) which maps injectively to H ° ( * ) Z 2 > < Z 2 , thus is isomorphic to a subring of H° (AT o(P , 2), JC*). The ring R(n) is actually a subring of T{U )/(vi,... n 0> 0 , v -i), 3n where (vi,... , i>3 _i) is the collection of regular functions giving the vector field V on UQ. n 12 Chapter 1. Introduction As in the case for schemes we can find a G we choose comes from a G M action on P The G action on Mo,o(P ,2). n M action M given by a matrix M ( A ) for A G G , such n M that this action together with the A action above is an (A , G )-pair (see Definition 7). 1 1 m We have an induced G action on Mo o(P , 2). In fact the cell U is the big BB-minus n M Q t cell (see [4]) of the unique A fixed point corresponding to a pair (C, / ) with C re1 ducible (this point is also G M fixed). We have an induced G M action on UQ and thus on R(n). This gives a grading of R(n), so R(n) is realized as a subring of the graded ring H° ( M ) , ( P , 2), JC*) thus is a subring of the graded ring H* (M , (P , n 2), C). n 0 sing 0 0 The last step is to show that R(n) is actually isomorphic to H* (M (P , n ing 0fi 2),C) as a graded ring. Getzler and Pandharipande calculate the Poincare Polynomial of Mo,o(P , 2) n in [9], which equals the Hilbert polynomial ofthe ring H* (M (P , n ing 0fi 2), C). We calcu- late the Hilbert Polynomial of R(n) and show that these polynomials are equal, thus the rings are isomorphic. The cohomology ring R(n) is generated by three elements, two in degree 1 and one in degree 2. Chapter 4 of this thesis focuses on the geometric meaning of these generators. We show how one can calculate the Chern classes ofthe rank three bundle E = 7r*/*0(l) as elements of R(n). Thus we have the second representation for the cohomology ring given at the beginning of this introduction. 13 Chapter 2 Coordinate charts for A4o,o(P ?2) n In this chapter we construct a collection of affine varieties, {Ui}, each isomorphic to A " , 3 - 1 each equipped with an action of Z , the group with two elements, and each having an 2 associated etale morphism &:cWM>,o(P ,2). n We shall show that the induced morphism of stacks [U /Z ]^Mo o(P ,2) Ji-. n i 2 ! is an open immersion in each case, and that the collection maps surjectively to Mo o(P , n t We shall define an A 1 action on Mofi(P ,2), 2). which will have a one dimensional fixed n locus. Two members of our cover UQ and U\ will cover the fixed locus. These members of the cover shall be used to calculate the cohomology in Chapter 3. First the standard description of the moduli stacks .Mo,o(P , 2), and the associated coarse n moduli spaces M ,o(P",2) will be given (Section 2.1). Then some general theory of A 1 0 and G actions will be developed, on P , and via P" on .Mo,o(P™,2) in Sections 2.2 n m and 2.3. Section 2.4 will focus on constructing one member of the cover {Ui}. We will show how to construct the whole collection in Section 2.5. In Section 2.6 we find a subcollection which covers the fixed locus under the action of A , this sub-collection consists 1 of two cells we call UQ and U\. In Section 2.7 we find the vector field on UQ and U\ 14 Chapter 2. Coordinate charts for M ,o(P , 2) n 0 induced by the A action. 1 Throughout this thesis we shall take the complex numbers C to be the ground field. 2.1 What is M ,o(P ,2)? n 0 In this section we shall first describe the coarse moduli space M o ( P , 2). We shall then n 0> describe formally the stack Mo,o(P , n 2), and explain the relationship between the two. We first consider the open subset M o(P ,2) of the coarse moduli space Mo,o(P",2). n 0) A closed point in M o ( P , 2) is an isomorphism class of pairs (C, /) where C is isomorn 0) phic to P and / is a morphism of degree two from C to P . In other words if we identify 1 n C with P with coordinates < s, t >, then / is of the form 1 / ( < S, t >) = < / (S, 0 t), fi{ , t), . . . , f {s, t) > S n where the / ; are homogeneous polynomials of degree two in s and t which do not simultaneously vanish. The image of C in P n is either a non-singular conic or a line. We say that a pair (C, /) is isomorphic to a pair ( C , /') if there exists an isomor- phism r : C —> C such that f or = f. In other words if we identify both curves with P 1 we have a projective change of coordinates for P , r : P —>• P , relating / and / ' . 1 1 1 There is a compactification .due to Kontsevich [12], M o(P ,2)cM o(P ,2) n n 0 ! 0 ) where the points of M ,o(P ,2) consist of isomorphism classes ( C , / ) where C is either n 0 isomorphic to P or to a tree consisting of two copies of P meeting transversally, in the 1 1 first case / is as before, in the second case / , when restricted to each copy of P , is an 1 15 Chapter 2. Coordinate charts for A4o,o(P , 2) n isomorphism onto a line in P . Thus f(C) is either two lines in P" meeting transversally n or a single line in P n with a special point corresponding to the image of the 'crossing point' of C. The concept of isomorphism class is as before. There are certain points in M ,o(P ,2) which are special because they have automorn 0 phisms associated to them. r : C —» C such that for A n automorphism of the pair (C, /) is an isomorphism = f. We shall call a point which has non-trivial automor- phisms associated to it a stack point. We need to decide which points in M , o ( P , 2) are n 0 stack points and what automorphism groups are associated to them. Lemma 1 The closed points z / M , o ( P , 2 ) with non-trivial automorphism groups are n 0 follows: (i) Points ( C , / ) where C is isomorphic to two copies of P meeting transversally, and 1 f maps both copies isomorphically onto the same line in P . 41 (ii) Points (C, f) where C is isomorphic to P and f constitutes a double covering of a 1 line in P . 71 In either case the automorphism group is Z . 2 Proof First suppose (C, f) is a class in M o(P ,2). So we identify C with P n 0( 1 as before and assume / is given by a series of n + 1 homogeneous polynomials. Suppose the image of C is not contained in a line in P . Then (C, f) can not be a stack point. This is the case n since the map / from C to / ( C ) is an isomorphism. Suppose ( C , / ) is a class in M (P ,2) n 0fi and / ( C ) is contained in a line in P . Then n 16 Chapter 2. Coordinate charts for Mo,o{P , 2) n (C, /) constitutes a two sheeted branch covering of the line and we have two branch points. First we observe that any automorphism must fix the ramification points. Suppose pi and p are distinct points in P such that f(pi) = 1 2 such that f or = f then either r(pi) = pi or r(pi) = p . 2 f(p2)- If T is an isomorphism Thus since / is a local iso- morphism r must either be trivial or be the automorphism which swaps the sheets of the branch covering. Therefore we have Z as automorphism group at these points. 2 If (C, f) is such that C is a tree of two copies of P image of C in P 2 is either a line in P n 1 meeting transversally then the or a pair of lines meeting transversally. In the second case we clearly have no automorphisms. In the first case, we can identify each component of C with P in such a way that / restricted to either component is the same. 1 With this choice of representative for the point (C, /) we have a unique automorphism, i.e. the one which interchanges the two lines. This automorphism again has order two. So again we have Z as the automorphism group for these points. 2 • We now shall describe the algebraic stack Afo,o(P 2). But first some definitions. n 5 Let pairs (C, f) be as before, i.e. C is a (arithmetic) genus zero, projective, reduced, connected, (at worst) nodal curve, and / is a morphism from C to P n of degree two (i.e. such that f*[C] = 2H ~ where H G Ai(P ) is the hyperplane class) such that no n l n component of C is mapped to a point under / (this last condition is the so called stability condition and it ensures that the automorphism group of the pair (C, /) is finite), then (C, /) is called a degree two stable map to P . n Let X be a scheme over C. 17 Chapter 2. Coordinate charts for Mofi(P , 2) n Definition 1 defn-family of curves A family of curves genus zero C over is a flat, X projective map TT-.C^X such that each geometricfiberC X is a (arithmetic) genus zero, projective, reduced, con- nected, (at worst nodal) curve. Definition 2 A family of stable maps from genus zero curves over X to P consists 71 of the data (n : C ->• X, \x : C -> F) : (i) A family of genus zero curves ir: C —> X (ii) A morphism p, : C —> F such that \i restricted to any geometricfiberis a degree two stable map to F . 1 Definition 3 Two families of maps over X, (TT! :d -> :d -> F), (TT 2 : C -> X,p 2 : C -+ 2 2 F) are isomorphic if there exists a scheme isomorphism r : C i —> C 2 such that iti — ~K O T 2 and Hi = fj.2 o r. Definition 4 A4o,o(P ,2) is the contravariant lax functor from schemes to groupoids, n /vf o(P \2) : {Schemes/C) r 0> X whereMofi(P ,2)(X) n (Groupoids) -+ A 7 o ( P , 2 ) ( X ) n 0) is the groupoid with whose objects are families of stable maps from genus zero curves over X to F, and whose morphisms are isomorphisms between such families. 18 Chapter 2. Coordinate charts for Mo,o(P , 2) n We note that the functor is contravariant since if we have a map of schemes g : X\ —> X 2 then we get an induced map of groupoids g* : A V o ( P , 2){X ) M ^ P " , 2)(X ) n 2 X where g'(n :C^X ,fi:C^P ) = (: n 2 where X Pl x 1 C -+ X ,p,op X2 l :X 2 x x C -+ P ) B X2 and p are the coordinate projections. 2 •Mofi(P , 2) is a lax functor since if we have a commutative of schemes n X -> Y \ I Z the corresponding diagram of groupoids is not commutative, however there is a natural transformation between the composition of functors M,,o(P ,2)(Z) -+ M> (P ,2)(Y) n M),o(P ,2)(X) n n l0 and the functor AT ,o(P ,2)(Z)->AT ,o(P ,2)(X). n n 0 Theorem 1 0 T/iere erciste a projective coarse moduli space M , o ( P , 2) associated to Mo,o(P' n 0 i.e. M , o ( P , 2 ) is a scheme and we have a natural transformation of functors n 0 4> • A V o ( P , 2 ) -> ^ o m n (*,M , (P ,2)) n 5 c / l 0 0 with the following properties: (i) (f>(Spec(C)) : M ,o(P ,2)(Spec{C)) n 0 -> Hom(Spec{C),M (P ,2)) n 0)0 tion. 19 is a set bijec- Chapter 2. Coordinate charts for Mo,o(P , 2) n (ii) If Z is a scheme and I/J : M.o,o(P , 2) —> Z is a natural transformation of functors, n then there exists a unique morphism of schemes 7 such that ip^jcxf). :M (P ,2)^Z n 0 ) 0 Where j : Hom(*,M (P ,2)) n ofi -> Uom(*,Z). (iii) The dimension o / M , o ( P , 2 ) is 3n — 1. n 0 (iv) JM ,O(P",2) is smooth. 0 Proof: [8, Theorem 1]. Theorem 2 in Fulton and Pandharipande's paper [8, Theorem 2] states that M 0 i 0 ( P " , 2) is locally a quotient of a non-singular variety by a finite group. We shall show in Section 2.4 that M ( P , 2) can be represented locally as the quotient of A ~ n 0 ) 0 3n l modulo an action of Z . 2 2.2 2.2.1 G and A actions on P 1 m G m Definition 5 and A 1 n actions An Algebraic Group is a variety G, together with a morphism p:GxG^G such that (G, p) is a group. G and A 1 m are examples of algebraic groups. As a variety, G m numbers C*. The morphism p is defined by p(Xi, A ) = AiA 2 20 2 is the non-zero complex Chapter 2. Coordinate charts for M ,o(P , 2) n 0 for Ai and \ 2 in C*. As a variety A 1 is the complex numbers C. In this case the morphism \x is defined by p,{ti,t ) = h +t 2 2 for ti and t in C. 2 Let G be an arbitrary algebraic group. Definition 6 An action of G on a variety X is a morphism 4> '• G x X —> X such that eachfixedg G G induces an isomorphism <j)(g) from X to X such that <f>(g){x) := <f)(g,x). These induced morphisms obey the group law, i.e. <t>{gi)4>{g2) = fiigm) ^(1G) = Ix where Ix is the identity morphism from X to X. 2.2.2 Two special actions on P n In our analysis we are concerned exclusively with A and G 1 m duced actions on Mofi(P ,2). The group P G L n on P n n + thus we can define an action of A or G 1 m algebraic groups from respectively A or G 1 m 21 actions on P i (PGL i = GL n + on P n to P G L i . n + n + 1 n and the in- / C * ) acts naturally by constructing a morphism of Chapter 2. Coordinate charts for M 0,0 (P , 2) n Consider the following morphism of algebraic groups from A to P G L i : 1 n + 1 t -»• M(t) = -1 t § 3! 0 1 t £ 0 0 1 (n- -2)! t n\ n-l (n-l)! t -3 (n- -3)! (n-2)! n (n- -1)! * -2 n 2! t n-2 n t 0 0 0 0 1 \ 0 0 0 0 0 \ t i.e. if we let 0 1 0 0 0 \ 0 0 1 0 0 0 ... 0 0 1 0 0 ... 0 0 0 1 0 0 0 0 A— V0 then M(t) = e . At It is easy to see that M is a morphism of algebraic groups, since if t\ and t are in A 1 2 then M't )M't ) = 1 M't +t2). 2 l We can also construct a morphism of algebraic groups from G m A A -»• 7V(A) = to P G L 0 0 0 ... 0 0 0 A 0 0 ... 0 0 0 0 A° 0 ... 0 0 0 0 0 0 0 0 0 0 Qo Q 1 22 2 A "0 0 1 A " Q \ n + i given by Chapter 2. Coordinate charts for Mo,o(P , 2) n where the on are integers. Without loss of generality we can assume that n = 0 i=0 in other words our matrix has determinant 1. We observe that for Ai and A in G. 2 N(X )N(X ) l 2 Thus this is indeed a morphism of algebraic groups. The notion of an (A , G )-pair shall be useful to us, [5, Chapter 1, Section 5]. 1 m Definition 7 Let G and A act on X via morphisms (j) and if) respectively. Then (tp, <f>) 1 m are called an (A , G )-pair 1 m if there exists a positive integer k such that for any t £ A 1 and A 6 G m 0(A)^(*)0(A- ) = ^(A**). 1 Suppose we require (M, /V) to be an (A , G )-pair (associated to which is a fixed integer 1 m k) then we must have N(X)M(t)N(\- ) 1 L e m m a 2 Suppose (M,N) is a (A , G )-pair. 1 m = M{\H). Let k be any fixed positive even integer, then the entries of N are determined by k and n, as follows: a 0 — ^ = nk 2 ctQ — ik, for i = 1,... , n. 23 Chapter 2. Coordinate charts for Moft(P , 2) n Thus in the case where n is odd I A, kn2 0 0 0 0 0 At 0 0 0 \-\ 0 0 0 0 0 0 , 0 N(X) V 0 k(n-2) X^ir 2 A 2 0 and in the case where n is even we have / A^ N(X) V 0 0 0 0 0 A^iH0 0 0 o At 0 0 o 0 1 0 0 0 0 A"" 0 0 0 0 0 0 0 0 . k(n-2) A k(n-2) Proof If we consider the equation rV(A)M(t)JV(A ) = _1 24 M(X t), k 2 ^ Chapter 2. Coordinate charts for Mo,o{P , 2) n the matrix on the left hand side is \ jn-2 +n-2 t~ (£=3)T n 3 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 v ° *A Q1 {n-2)\ ai—a„ (n-l)! A *i f\Ctn-2-<Xn-l \On-2-a„ 2! 1 and the matrix on the right is I 1 t\ *"- %\ k t"\fc(n-l) (n^l)! 1 A t~ n 0 \ \k(n-2) 2 2k \k(n-3) 3 t"~ \fc(n-2) (rtT2)! 2 A t"- \fc(n-l) (^T)! 1 A 0 0 0 1 tx 0 0 0 0 1 tx 0 0 0 0 0 1 k k Equating we see that k — O>Q — ct\ = OSQ — ai 2k nk = ao — a . n So in particular for each i — 1... n, on — CXQ — ki. Now n o = = i=0 a + (a - k) + (a - 2k) + ... + (a - nk) = , ^ . fn(n + l)\ (n-H)ao-fc^ J- Q Q 0 V Q J 2 25 Chapter 2. Coordinate charts for Mofi(P , 2) n Therefore nk *o = as required. 2.2.3 Y • A BB-minus decomposition of P Suppose we consider the G m n action given by N on P". We will not specify a value for k (except for requiring that it must be even) since all our results are independent of k. We can decompose P into locally closed subsets using this action. In general we have the n following result due to Bialynicki-Birula [4]: Theorem 2 Let G m points are isolated. act on a projective non-singular scheme X such that all the fixed Then X can be decomposed into locally closed subschemes each of which is isomorphic to a vector space and contains exactly onefixedpoint. One canonical decomposition satisfying the requirements above is the Bialynicki-Birula minus, or BB-minus decomposition. Definition 8 Suppose G acts on P , with isolatedfixedpoints Fo,... ,F . 71 m n The V BB-minus cell C{ is d = {xe P*\ lim \.x = Fi\. A—>oo Before we can calculate the BB-minus decomposition of P find the fixed points under this action. 26 n under the action N we must th Chapter 2. Coordinate charts for A4 o(P , 2) n 0l Lemma 3 The fixed points under the action given by the matrix N are Fi =< 0,... , 0, 1 0,... , 0 > P position h for % = 0 , . . . , n. Proof Suppose x —< x , XI, ... ,x 0 > is fixed, then at least one of the Xi is non-zero. Suppose n Xj is non-zero. Then X"? i Xj = p,Xj for some p, ^ 0. Thus p, = X^~ . k jk addition xi ^ 0 also for I ^ j, then p = X r~ , ! Suppose in which is impossible. Thus we can lk assume x\ = 0 for / ^ j, and Xj = 1. Clearly j can take on any value from zero to n. Thus for any even positive integer k, the G m Lemma 4 Let G act on P via N then 71 m Ci = {< x ,Xi, ...x >e F^lxj = 0 for j <i k i = 1}, n 0 for i — 0,... action has fixed points F{ for i = 0 , . . . , n. • r ,n. Proof Let us first consider C which is the set of points in P which move to the fixed point n 0 F =< 1,0,... , 0 > as A 0 oo. Now if x =< x , x ,... 0 x ,x n > is in C we clearly must 0 have x ^ 0. If x = 1 we have 0 0 lim (A. < l , x i , . . . ,x n = lim (< 1, X~ x X~ x ,... k , X~ x 2k u nk 2 n >) >) =< 1,0,... , 0 > Thus Co = { < a r , x i , . . . , a ; > G P | a ; o ^ O } ^ ^ . n 0 n 27 n Chapter 2. Coordinate charts for A^o,o(P , 2) n The set C\ C C' — {< XQ,X , 0 ... x x n > \x = 0}. Using the same type of argument as Q above we see that Ci = {< x ,x ...x 0 lt >e P \x = 0 & x = 1} £ A ~ . n n n 0 l x Continuing in this way we find that Ci = {< x ,Xi,.. 0 .x >e P \xi = 0 for i = 0 , . . . ,i - 1 & x = 1} = A ~\ n n n t as required. • We call the cell Co the big cell (because it has highest dimension). It is special because it contains the unique A fixed point F . 1 0 2.3 Gm and A actions on JW ,o(P , 2) 1 n 0 Let G be an arbitrary algebraic group acting on P . We can describe an induced action n M o ( P , 2 ) as follows: n 0> Definition 9 Let (C, / ) be a point in M , o ( P , 2) then for g G G, we define g.(C, f) to n 0 be g.(CJ) = (CM9)°f) where <p(g) is the isomorphism on P induced by g. 71 This action is well defined since all representatives of the class (C, / ) have the same image in P . One can define a compatible action on the stack Mofl(P ,2) as in [10]. n n Let A act on M ,o(P",2) via the action on P 1 0 the fixed locus under this action. 28 n given by M. We would like to find Chapter 2. Lemma 5 Coordinate charts for Mo,o(P , 2) n Thefixedlocus under the action given by M on M , o ( P , 2) is parameterized n 0 by P and is as follows: 1 • Points ( C , / ) in M ,o(P ,2) where f(C) is a conic in P given by the ideal n 71 0 I c for c G A . 1 = (Xl We shall use F ^ — 2XQX + cX\, 2 1 Xz, X t , . . . , X) n to denote these points. • The point (C, f) where C is isomorphic to two copies of P intersecting transversally 1 and f maps each copy of P isomorphically onto the line given by the ideal 1 loo — (X , X3,... 2 , X) n and the crossing point maps to the unique A fixed point in P 1 71 namely the point FQ =< 1,0,... , 0 >. We shall use F^ to denote this point. Proof If a point (C, f) is fixed then for each t G A , f(C)=(M(t) o f)(C) 1 as a sub-variety of P , thus f[C) must be invariant under the action of A . For (C, /) G M , o ( P , 2), / ( C ) n 1 n 0 is either a line or a conic in P (the conic may be degenerate). Thus we are looking for n conies or lines in P which are invariant under the action of A . n 1 Let Vi = {< x ,xi,... 0 , x > G P" \ Xj =0 for j > i}, n for i = 0,... , n, then it is easy to see that the subvarieties Vi are invariant under the action of A and 1 Vi = P \ If we let x eVi-Vi-i then the orbit of x under A is a curve of degree i. Thus 1 all our invariant points (C, /) have the property that / ( C ) is contained in V . 2 The A 1 action restricted to V is given by the following matrix (when we identify V 2 2 29 Chapter 2. Coordinate charts for M o ( P , 2) n 0] with P ) : 2 fi t 2 0 1 One can easily verify that the equation of the conic parameterized by the points above is Xl - 2X X 0 2 + cX\ = 0, where c = 2x -x\. Q These conies are all nonsingular and we have a one-to-one correspondence between these conies and points c e A . Thus if ( C , / ) is such that / ( C ) lies in V , and if when we 1 2 identify V with P , / ( C ) is one of the conies above, then (C, / ) is a fixed point under 2 2 the action of A on M , o ( P , 2). 1 n 0 What about fixed points (C, / ) in M ,o(P ,2) where / ( C ) is a line or a pair of disn 0 tinct lines in P ? Clearly since we have only one A invariant line in P , (namely the n 1 n subvariety Vi) the latter is impossible. We observed above that any degree one invariant curve lies in Vi, so / ( C ) must lie in Vi and be the line X = 0 (identifying V with P as 2 2 2 before and viewing Vi as a subvariety of V2). We note that if (C, f)=(C, M(t) o / ) then any 'special' points on / ( C ) must be A invariant, by special points we mean branch 1 points or the image of the 'crossing point' if C happens to be reducible. Thus since A 1 has only one fixed point / cannot constitute a branch covering. We conclude therefore that the only possibility is that C is reducible and / ( C ) is the line X 2 = 0 and the crossing point maps to the point < 1,0,0 >. Translating this information to P n (and letting Xi be the coordinate functions in P 30 ra Chapter 2. Coordinate charts for Mo o(P , 2) n t for i = 0 , . . . , n) the fixed locus is claimed. • 2.4 An open substack of A^o,o(P ? 2) n In this section we shall construct a morphism <j> from the affine space A ~ 3n viewed as a x stack, to jMo,o(P ) 2). We shall show that the image is an open substack U of Mo o(P , 2), n n t and that <f> is etale. In fact we can find an action of Z on A such that the induced 371-1 2 map from the quotient stack 0: [A "-7Z ] ^M,,o(P ,2) 3 n 2 is an open immersion. A map of stacks from A ' to A^o,o(P", 2) is given by a pair 371 1 A ~\ f : C —> P ) € M,,o(P , 2 ) ( ^ " - ) . (TT : C Theorem 3 Let C =P 3n L n x A~ 3n n 3 with projection n to A ~ l 3n 1 and let f be the rational l map from C to P defined as follows: 71 /(< s,t >, (m,6i,... A i ^.n 2,2,-- - ,r ,i> ».2)) r r n =< st, ms + bist + t , mr ,is 2 2 2 2 + b st + r t ,... 2 2 2j2 , mr s + b st + r t 2 njl 2 n n>2 >. Let C be the blowup of C along the subvariety where m = t = 0, and let f be the induced morphism from C to P . Then the data 71 c -UP 71 7f J,, 31 Chapter 2. Coordinate charts for Mo,o(P , 2) n gives a morphism of stacks 4>: A - 4>{X): A*"-\X) 3 -+ 1 AT ,o(P ,2), n 0 -> M (P ,2)(X), n 0<0 where if a G Homs h(X, A " ) then (f)(X)(a) is given by the following data: 3 - 1 c x 3n-i X C -> A C 1 i f 1 7T Proof We have to show that after blowing up we have a family of stable maps. The rational map / is clearly defined when m and t are not both zero since st and ms + b\st +1 2 2 do not simultaneously vanish. Let U and U be respectively the subvarieties corresponding t s to the points where t ^ 0, and s ^ 0, so we have U = U — A . 3n s m,b ... , b r ,i, r 2 ) 2 (|, m,b ... , b , r ,i, r 2 ) 2 u m 2 t , •• • ,r n>u U has coordinates s r ^j , n>2 and U has coordinates t u n 2 , . . . , r ,i, r , ) • n n 2 We construct the blow-up, U , of U along the subvariety V(t/s,m) where the map / is s s undefined. Let x £ U and let s Vs = | ( x , < p , 9 >) G tf, x P | mq = 1 U can be covered by two affines U and U both isomorphic to A s p q U = p • 3 n where {yeU \p^0} s with coordinates -, m, h,... vP ,b, r n 2)1 , r , , ••• ,r , r 2 2 nA n>2 / 32 ) where m ( - ) = - , VP/ S Chapter 2. Coordinate charts for A i , o ( P , 2) n 0 and U = {yeU \q^0] q a with coordinates 0 , ^, 6i,... , b , r ,i, r 2 n 2 , , . . . , r„,i, r ^ 2 where n> Then / induces a morphism / as follows: / : U -)• P n t m, 6i,... , b , r ,i, r , , • • • , r„,i, r „ ) / n 2 2 / :U P" p / P 1 -, m, bi, • • • , K, r , i , r 2 + &i ( - ) + m ( - j \Pj 9 + 6i + - , r 2 > 1 s n n > 2 \PJ 2 P n PA 1 \ ,... , r ,i, r n q ft 2 > 2 , r„,i + b ( - ) + mr„, \pj f:U ^ = (l,- )2 /< .? \ 2 S -i 2 ( - ) + 6 + r , ( -s ) , . . . ,r„,i ( - ) +6 V?/ V / W 2 2 + n 2 These glue to give a morphism from C = ^ l " 3 1 x P to P . 1 n We now need to show that (# : C ->• A ~\ 3n f : C - ) • P ) e X o , o ( P , 2)( 4 " ) ] n n 3n J It is easy to see that the morphism ?f : C ->• A3 n - l 33 1 Chapter 2. Coordinate charts for Ato,o(P , 2) n is flat. The fiber C above x = (m, bi,... , r x ) is isomorphic to P in the case m ^ 0, 1 n;2 and is isomorphic to {< s,t > x < p , >G P x P | tp = 0} 1 1 g in the case m = 0, so in either case we have a projective, arithmetic genus zero, (at worst) nodal, reduced, connected curve. The induced morphism from the fiber is of degree two in either case. • The next lemma gives us more information about the images of geometric points under the map <j>. Specifically if a G A ~ Zn object (a - * A ~ ) 3n Lemma 6 fiber of in P 11 C l G A ~ (Spec(C)) Zn l we would like to know what is the image of the under <f>. l Let a — (m, 6 ,... , b , r , i , r ,2, • • • , r i, r X n over a. Then ifm^O, 2 n> 2 and r^i / T*J ) 2 ) GA~ Zn n>2 l be fixed. Let C be the a for some i = 2,... , n, f(C ) a is a conic contained in the unique plane containing the points < 0,1, r i , . . . , r 2 > If m — 0, then C n > 1 >, < 0,1, r ,2,... , r„, >, < l,b 2 2 u . . . ,b n > . isomorphic to two copies of P meeting transversally and f(C ) 1 a a pair of (not necessarily distinct) lines, one through the pair of points < 1, &i, . . . , & „ > , < 0. U 2,i, • • • , r ,i > r n and one through < 1,6i,... , bn >, < 0,1, r , , . . . , r , > . 2 2 n In this case, the crossing point of C maps to < 1, b\,... ,b 2 a n 34 > under f. is a Chapter 2. Coordinate charts for Mo,o(P , 2) n If m ^ 0 and r^i = r i ] 2 for i = 2 , . . . , n then f : C -4 P is a double covering of n a the line through < 1,b\,... ,b > and < 0 , 1 , r i , . . . , r i >. n 2> n > Proof Clear from the definition of / , and C. • L e m m a 7 Let Z act on A ' 371 1 2 (m, bi,... , b , r i , r n 2 j 2 ) 2 as follows: , . . . , r„,i, r n>2 ) ->• (m, & i , . . . , b , r n 2>2 , r , i , . . . , r„ 2 |2> r i). n> TVien we /iaue an induced action on C suc/i r7ia£ 7r and f are equivariant, and in particular forpeZ , 2 f(pc) = /(c) force C. Proof Let a: be the non-identity element in Z . Then a defines a rational map 2 a : P x A " " —» P x A 1 3 1 1 a(< s, t >, (m, 6 i , . . . , bn, r i , r 2> 2 | 2 3 n _ 1 , . . . , r , i , r„ )) n l2 =< t, ms >, (m, 6 i , . . . , bn, r , r i , . . . , r , , r„,i)), 2j2 2 | n 2 which is defined when m and t are not both zero. Clearly a = 0. 2 One can readily verify that n (the projection TT : C -> A 3 n _ 1 ) is equivariant with re- spect to this action. We also have /(<*(< s, t >, (ra, 6 i , . . . , bn, r ,i, r , , . . . , r , i , r„, ))) 2 = / ( < i , ms >, (m, 2 2 n &„, r , , r , i , . . . , r 2 35 2 2 2 n>2 , r„,i)) Chapter 2. Coordinate charts for Mo,o(P , 2) n =< mst,mt +bimst+m s ,mr2, t +mb st+r im s ,... 2 2 2 2 2 2 2 ,mr t +rnb st+r im s 2 2 2) =< st, ms + bist + t , rar ,is + b st + r t ,... 2 2 2 2 , mr 2 2 = /((< s, t >, (m, 61,... 2 nj2 2>2 , bn, r 2 n > s + b st + r t > 2 n>1 2 nj 2 n n>2 , i , r ,2, ••• , r„,i, r„, ))). 2 2 So / is equivariant with respect to this action. When we blow up along the subvariety m = t = 0 we have a morphism a from C to P x A ~\ 1 3n C I pi a \ pi 3n-1 x A x 3n-1 A Blowing up again we have an isomorphism a from C to C, and the following diagram commutes: C C a I 4- pl x pi 3n-1 A 3n-1 x A Clearly / o a — f by our construction, so / is equivariant with respect to this action. • We now look at <j>(A ~ ) as a substack of A T , ( P , 2). 3n l n 0 36 0 Chapter 2. Coordinate charts for Mo,o(P , 2) n Theorem 4 Let U be the stack defined as follows: U : (Schemes/C) —> X -¥ (Groupoids) U(X) Where U(X) C .Mo,o(P , 2)(X) is a subgroupoid of stable maps given by the data: n (v.C^XJ-.C^P"), (where this data represents an element of Mo,o(P , 2)(X)) satisfying the following: n Consider f*(Xi) 6 r ( C , / * 0 ( l ) ) where the Xi are the usual coordinate functions on P , 71 and let D = Z(f*(Xi)) for i = 0,1 then { (i) The induced map 7 r : D —>• X is unramified, 0 (ii) The divisors D and D\ do not intersect. Q Then U is an open substack and (j): A ™* —> A^o,o(P , 2) factors through U. 3 1 n Proof First we must verify that these conditions indeed define an open substack of A fo,o(P 2). / n 5 Fix a scheme X and an element (7r:C->X,/:C->P ), n of M),o(P ,2)(.X). Then it suffices to show that if x e X is such that (C ,f) satisfies n x the conditions above, then we can find an open set x 6 U C X such that the family C restricted to U satisfies the conditions also. Certainly the locus of ramification points on D , R, is closed in D and the locus where Q 0 D and Di intersect is closed also. So U = X - ir((D n L>i) U R) is open and contains x. 0 Q 37 Coordinate charts for Ato,o(P > 2) Chapter 2. n To show that the image of A 3 n _ 1 is contained in the substack U it suffices to show that the fibers of the morphism / from C to P" satisfy the two conditions above. This is equivalent to showing that for x G A 3 n - 1 , the image of the fiber C under / is contained x in Co U C i (where the Cj are the BB-minus cells of the action of N on G , see Lemma 4) m and exactly two points in C have their images in the hyperplane X — 0. We shall 0 x consider two cases, m = 0 and m ^ 0: If m ^ 0 then C is identified with P with homogeneous coordinates s and t, and / 1 x restricted to C is given by x /(< 8,t>) = < st, ms + bist +1 , mr ,is + b st + r t ,... , mr„ s + b st + r , t > . 2 2 2 2 2 2 2 2>2 2 ;1 n n 2 We notice also that the image has the property that the first two coordinates do not simultaneously vanish thus the image is contained in Co U C\, and exactly two points map to the hyperplane XQ = 0 namely < 1,0 > and < 0,1 >. When m = 0 the fiber C can be identified with x C = {< s, t > x < p, q > \tp = 0} and / restricted to C is given by x /(< s,t> x <p,q>) - (sq, sp + hsq + tq, r sp + b sq + r tq,... , r isp + b sq + r tq). 2A 2 2>2 n> n nfi One sees easily that the points (< 0,1 >, < 0,1 >) and (< 1,0 >, < 1,0 >) map to the hyperplane X = 0, and the image is contained in Co U C\ as required. • 0 Notation Convention 38 Chapter 2. Coordinate charts for A^o,o(P ) 2) n From now on we shall use / in place of / and C in place of C and a in place of a, and we let Uo = A ~ Zn l i.e. the data C -4 P n Uo will represent our chosen map of stacks ^:^ -^lMo,o(P ,2), n 0 and we have an action of Z on C given by a, and with respect to this action / and n 2 are equivariant. Lemma 8 The induced map <f>: A ~ 3n l -)• U is surjective. To prove this we shall need the following lemma: Lemma 9 Let (C,g) € U(Spec(C)), then if (C,g) is a stack point there is exactly one point p e U such that (C ,f) 0 p is isomorphic to (C,g), otherwise we have two such points, and these points lie in the same orbit under the action on Z . 2 Proof First let us assume that C = P . In this case g is represented by 1 g :< s,t>->< g (s,t),gi(s,t),... 0 ,g (s,t) >, n where the & are homogeneous of degree two and do not simultaneously vanish. 39 Chapter 2. Coordinate charts for Ato,o(P > 2) n Since (C, g) G U(Spec(C)) we have two distinct points of C mapping to the hyperplane Xo = 0 and no points mapping to the intersection of this hyperplane with the hyperplane X\ = 0. Thus g has two distinct roots, so we may assume #0 = st. Since 0 go and g cannot vanish simultaneously we must have g = a i ^ s + a^st + ai i 2 x fli.i) i , 3 a x 0- If w e with 2 ; 3 compose G with the projective change of coordinates r for P given 1 by the matrix 1 0 0 — 11,3 we have that (C, J O T ) is isomorphic to (C,g) and gor < fll,3 o CLi 2 ^1 3 9 st, a is H -st + —j-t,.. x ' < ai,3 3 + ai st + t ,... , a , i O i , 3 S + a st + = ( st,a iai s 2 h 1 is given by 2 t3 ^t C 2 n )2 n<2 \ Q 2 3,l We shall use this representative for the isomorphism class (C, g). In this case the image is contained in the linear subspace of P spanned by n < 1, O i , • • • , a , >, ( 0, 1, — , . . . , — n 2 ) 2 If this is a plane, then \ Ol.l ^ ^ 01,1 ' > , ( 0, 1, — S . • • , — Oi,i/ \ Oi Oi | 3 > 3 for some i = 2,... , n. In this case there are exactly ai,3 two points pi and p in C/o such that (C , f) is isomorphic to (C, g) namely 2 Pi ^2,3 2,l a ^ 1 , l l , 3 , ^1,2; • • • ) Q.„,2, , a °n,3 n,l a , ••• j ^1,3 , a °1,3 l,l and fll,lGl,3> ^1,2, • • • , Qn,2) G2 Q )3 Ol,3 , 2,i 0,1,1 o„ , ••• > i 3 Ol,3 j a ,i\ n I • 01,1/ If the subspace spanned by the points above is a line then ^ —^ iox i = 2,... , n, so we have exactly one point p in U such that (Cj,, / ) is isomorphic to (C, g) namely 0 p — I a i ia , a i 2, • • • , a , 1 x 3 n2 ^2,1 , °2,1 a i . i 01,1 40 n,l a oi,i , n,l a ai,i Chapter 2. Coordinate charts for Mo,o(P , 2) n In the case where C is reducible, we can assume that C = {< s,t > x < p,q >G P x P 1 and g : C —• P n ^ = 0} 1 is given by g{< s,t> x <p,q >) =< g (< s,t> x < p, q >),..., g {< s,t> x <p,q>) > 0 n =< a ,ips + a qs + a tq,... , a ips + a qs + a tq > . 0 ofi 0t3 n> nfi n>3 Since (C, g) G U(Spec(C)) we can assume that g vanishes at the two distinct points 0 (< 1,0>,< 1,0>),(< 0,1 > , < 0 , 1 >), therefore we can assume that go = qs, and since g and g\ do not vanish simultaneously, 0 01,1,01,3 0. Composing g with the isomorphism r' : C C given by T' :< s,t > x <p,q >-> (s,—t) x (—p,q) \ ai / \oi,i / >3 g or' :< s,t > x <p,q> &21 ^2 3 1 ^n3 \ gs,ps + o,\ oqs + tq, —*-ps + a , gs H -tq,... , —-ps + a , gs H ai,i ai ai,i ai, / in this case the image is two (not necessarily distinct) lines, one through the points 2 2 n ) 3 i /n i ° 2 , 1 < 1, ai,2, • • • , a ,2 n ( 0,1, >, \ 3 n,l a ,• • • , Oi,i Oi,i and one through the points < l,ai,2, • • • ,a ,2 >, ( 0,1, n 41 a ,3 2 ai,3 2 a 3 ni ai ) 3 ), Chapter 2. Coordinate charts for Mo,o(P , 2) n If these lines are not distinct (C, g) is a stack point. If the lines are distinct, we have exactly two points p\ and p in UQ such that (C , f) is 2 Pi isomorphic to (C,g), namely and • Proof of Lemma 8: To show that a map of stacks is surjective it suffices to show surjectivity for geometric points (see [2]). Therefore we must show that if (C, g) G U(Spec(C)), point p e Uo such that (C x p,fop ) Uo x we can find a = (C , f) is isomorphic to (C, g). This follows p immediately from Lemma 9. • Theorem 5 We have an induced map of stacks <t>: [A - /Z ] 3n l 2 -»• U which is an open immersion. Proof We have shown at this stage that the image is U. Thus to prove this theorem we need to 42 Chapter 2. Coordinate charts for Mofi(P , 2) n show that the map <f> is etale (Lemma 12), radiciel, and representable (Lemma 13), then, since the image is the open substack U of .Mo,o(P >2), the map is an open immersion n (see [2]). • First we show that <f> is etale. From Deformation Theory we have the following lemma: Lemma 10 Let X be a scheme over C. A morphism tp iP:X^M (P ,2) n 0fi given by the data C -4 P n I X is etale at a 6X if and only if the induced map H°(C ,T )^H°(C ,rT^) a c a is bijective. Lemma 11 <j> is etale. Proof By Lemma 10 above it suffices to show that both H°(C ,f*Tpn) a and H°{C ,T ) a c are isomorphic to C " " , and that the induced linear map between them is bijective. 3 4 2 Let a = ( m , 6 i , . . . , b , r , i , r n 2 2 f 2 ,. •• ,r ,i,r , ). n n 2 when m ^ O and when m — 0. 43 We shall we shall deal with two cases, Chapter 2. Coordinate charts for A^o,o(P , 2) n Let U = {< x ,x ... Xi Since / ( C ) C U Xo UU 0 ,x >E P \xi ^ 0}. n u n we need only consider the tangent bundle for P , Tpn, on these n Xl neighborhoods. Suppose we use affine coordinates ^ for i ^ j on U , and list them as follows: Xj XQ Xj—^ Xj Xj^.\ Xn Xj Xj Xj Xj > • • • ) We choose local trivializations for Q,pn in the standard way, i.e. &:fi »|tfx P 4 (0 T ^ Uxi d[ Xj (0,... ,0, x, If we let it / V Suppose we let $ j then in particular <f) j — <j>i o -if ^ ^0X2 XQX$ xf xj 0 ,i o , 0 , . . . ,0), position th is given by the matrix: \ x\ 0 XQ Xl 0 0 0 0 XQ 0 Xl 0 0 XQ Xl 0 0 0 0 XQ Xl J be the locally trivializing maps for the dual sheaf 7p" then transpose of the matrix above. Suppose first that m ^ 0. Consider the following diagram: P = 1 C C 4 i a 44 -4 P n 4>\ fi is the Chapter 2. Coordinate charts for A^o,o(P , 2) n Suppose we give P coordinates < s, t >, then the induced map from C to P is given 1 n a by < s,t >—K st, ms + bist +1 , mr2,\S + b2st + r ^t ,..., rar is + b st + r ^t > • 2 2 2 2 2 2 ni 2 n n = < / o ( M ) , / l (*,*),••• Jn(s,t) > The trivial neighborhoods Ui pull back to the trivial neighborhoods £//., where U/ — {< { s,t ><= P\fi{s,t) ^ 0}. In particular U = P fo 1 - ^ ) and U = P -V{ms +b st+t ), 1 h 2 2 x and U U U = P . 1 fo h The gluing matrix <^ pulls back to 0 I t 2 foh ft 0 0 0 fo 0 h o Jl 0 fl /o/n-1 0 fo fl 0 ft h I An element of H°(C , / * T p » ) is a global section of / * T p " , so is given by a on Uf. for j = 0,1, where g j is a homogeneous polynomial of degree four is s and t, and it M i ( g i ) = goj0 This forces —<7i,i — gift and + /oft,i = /i0«,o for % = 2,... , n, /o<7i,l = /l<7i,0 45 — /j<7l,0- Chapter 2. Coordinate charts for Mo,o(P , 2) n Suppose we let gij = a^s + bijsH + . . . + e^t , then since g \ = —<7i,o we have in 4 4 it particular a± i = —ai,o and e^i = —ei , and the equation above becomes t i0 st(a s + ... + e t ) A 4 iA iA = (ms + ftist + t )(a s* + ... + e t ) - ( m r ^ s + M + r t )(ai s 2 2 4 ifi 2 2 ii0 it2 + ... + e t ) 4 4 fi li0 Taking s = 0 we have 0 =& taking t = 0 we have 0 = ma ifi - mr iai i} =>• r a fi itl — a. lfi ifi Thus the polynomials g^\ are entirely determined by the polynomials p; . The polynomi)0 als gifi, for % = 2,... , n , are not entirely free, their s and r coefficients are determined by 4 4 (71,0. Thus for i = 2 , . . . , n, the polynomials (?; have each exactly three free coefficients. )0 Altogether we have 5 + 3(n - 1) = 3n + 2 free coefficients. Thus H°{C , / * T » ) S C a 3 n + 2 P in the case m ^ O , and has a basis given on £// by 0 \ t dwi' t dwi' dwi' s dwi' s die i ' 2 2 where W; = We shall now show that H°(C , Tc) = C a 3 n + 2 also. The fiber above the point a is contained in Up U U since m ^ 0, so we need only consider T c restricted to these neighborhoods. t Recall that U has coordinates t and U has coordinates v -,m,6i, 46 Chapter 2. Coordinate charts for Mo,o(P , 2) n where m P Since m ^ O w e can replace £ with (^)(| )• We have locally trivializing maps for fic|r/ nt/t p given in the usual way: <j> :Sl \U ^(0 f n t c t Vt 4> :Sl \U ^(0 ) , 3n p and the gluing matrix c p 4> = <$> o <^ x pt p Ut is - M (S) / - (i) (!) l3n-l V If we allow U and U to denote the neighborhoods U and U pulled back to C (identified a t p t a with P ) , then the gluing data for 7 c pulled back to P is the transpose of the matrix 1 1 above i.e. if we let M denote the gluing matrix which is the pullback of 0 , then V st 4 M ( S ) -(i)(f) M st ••• 0 \ 0 = I3n-1 V A global section of TQ is given by a 3n-tuple of rational functions on each neighborhood, such that if we multiple the 3n-tuple of functions for U by M s of functions for U . t at we get the 3n-tuple Clearly the functions in positions 2 to 3n must be constant, say the constant functions A ; , . . . , k , 02,1,02,2, • • • , c ,i, c 2- If we let £ and & denote the 0 n n n> rational function in the first position for U and U respectively, we see that i = j = 2 s t and -<»> (?) H 3 (?)*-*• 47 Chapter 2. Coordinate charts for Mo,o(P , 2) n Therefore stk 0 9= mf. m Thus g is dependent on / , and / is a homogeneous polynomial of degree two. In particular if / = a s + aist + a t , then 2 2 0 2 g = —ma s + (—ma\ — — ^ st — ma t . m 2 2 0 2 Altogether we have 3n — 1 constant functions, and / has three free coefficients, so we have 3n + 2 in total, therefore H°(C , T ) = C a 3 n + 2 c in the case m # 0, and a basis for T c on Ut is d s_d d d_ d_ d * 3(f)'t9(f)'a(f)'am'-56i'-"'cV 2 n i 2 i /- We now need to show that the induced map H°(C ,T )^H°(C ,rTpr>) a c a is bijective. The universal map / from C to P to f~ (U ) induces a map from f*Qp* to tic- Restricting C U and using U coordinates, we have l XQ f N t t ((?*)) = (? (?) 4 ^ m +b (?) 4 - ' +mr +b • • •' +r2 2 mr ^ (?) A - ) +b +r 2 and so, (™ , -r (^y^ = 2 x ^ ( zoO l^"' " "' = 1 r 2 (s ) 2 2t2 ) ^ (?) + ^ 48 (?) d m + d b n + m (?) d r ™ + ( d f n 2- Chapter 2. Coordinate charts for M o(P , 2) n 0i Therefore the induced map from a c\f-iu /*fipn X 0 xo is given by the 3 n x n matrix ( mr ,i - r 2 f r (&) mr i - r 3> (f) 2 i l i 2)2 (£) (f) 0 0 o 0 0 0 0 1 0 0 0 0 1 0 0 0 m (!) s 0 0 0 0 0 m (!) 0 0 0 0 m (!) 0 V \ mr, (f) ^3,1 0 3 ) 2 s 0 0 0 0 0 0 0 0 m (!) 0 0 The transpose of this matrix gives the induced map from T c|/-ir/z 0 - > /*7p»|/-i% . 0 We need to write down the matrix representing the induced map H (C ,Tc)->H (C J*T n), 0 0 a a p with respect to our chosen bases. This matrix will be a (3n + 2) x (3n + 2) matrix. 49 Chapter 2. Coordinate charts for Mo,o(P , 2) N Let v = (a ,ai,a ,k ,... 0 2 , k , c ,i, c 0 n 2 , . •. ,0,1,1,(^,2) € C " " represent the section of 371 2)2 1 2 To, given on U by the 3n-tuple of functions s / aos +ai st+atf «2 2 2 \ A; 0 n,2 \ C J and Ut by ( (-mao)s +(-J-fl-TOOi)^t+(-ma2)t 2 i \ 2 2 *1 \ n,2 / C We multiply the U representative by the transpose of the matrix above to get the corret sponding element of H°(C , / * T p » ) . Letting a (-ma )s 2 0 + (-^ - mai)st + (-ma )t 2 2 9= t 2 we have / h (f) + k + (m-£)g ^ 1 (fc r ,i + 0 2 mc ,i) 2 (f) + k 2 +c 2i2 (f) + ( m r , i - r , 2 2 (fc r ,i + mc ) (f) +k + c , (f) + (mr ,i - r 0 \ 3 3>1 3 (f) +k (kor ,i + mcn^) n Writing the i th 3 n 2 +c n>2 3 (J) + (mr ,i n 3 ) 2 # 2 (j)) 5 - r , ( n 2 # / coordinate as a rational function with sH in the denominator, we have 2 the following numerator: -m a r^is 2 4 0 + (-rj m a + mc i)s t + (-m a rt,i + ma r 2 + ki)s t + 2 )1 3 1 it 2 2 2 50 0 it 2 Chapter 2. Coordinate charts for Mo,o(P , 2) n c i>2 + T + mai7-i,2 ) t s + 3 ^ z r ^ m ma ri 2 When j = l we retrieve the numerator by taking r^i = r i We identified an element of H°(C , / * T p « ) with C a 3 n + 2 j2 t . 4 t 2 = 1 and ci,i = C i = 0. )2 by listing the coefficients of the coordinate functions in increasing powers of t and ignoring the r and s 4 4 coefficients in positions 2 to n since they are not free. Thus the section above corresponds to the following element of C 3 n + 2 with respect to our chosen basis: \ —mao 2 —m ai 2 \ —m a + ma + ki 2 2 0 ma 2 + mc i —r ^m a\ l 2) 2 —m a r + mao^2,2 + k 2 2 2)1 2 C2,2 + ^ + ™ r2,2 ai -r im ai 4- mc i 2 3i 3j -m a r i + ma r 2 2 3! 0 3)2 +k 3 +-^r+ "W3,2 r c ,2 3 -r im ai + mcn,i l nt - m a r i -I- ma r 2 2 2 n) 0 51 n> + k n Chapter 2. Coordinate charts for At o,o(P , 2) n Therefore our map is given by the matrix: ( a 0 ai a k 2 0 kx .. . ki . Ci,l C i s —mr 0 0 0 0 .. . 0 . . 0 st 0 —mr 0 0 0 .. . 0 . . 0 st m 0 —m 0 1 ... 0 . . 0 st 0 m 0 m 0 .. . 0 . . 0 t 0 0 m 0 0 .. . 0 . . 0 0 0 0 -m r 0 0 0 . . 0 .. m 0 0 -m r 0 0 . . 1 .. . 0 0 r.,2 0 .. . 0 ... 0 1 4 9 3 2 2 3 4 st 3 st 2 st 3 2 0 2 2 iA mn,2 2 iA 0 m 52 i 2 Chapter 2. Coordinate charts for Mo (P , 2) n t0 where the three rows indexed by i occur for % - 2,... , n. This matrix is invertible for all n > 2, with inverse I 1_ m 2 0 1_ M^: m Q 0 0 0 0 0 0 .. . 0 0 0 .. . 0 0 0\ 0 0 0 0 0 0 .. . 0 0 0 .. . 0 0 0 0 m 0 0 0 0 m 0 0 0 .. . 0 0 0 .. . 0 0 0 0 1 0 m 0 0 0 0 .. . 0 0 0 .. . 0 0 0 \_ 0 1 0 m 0 0 0 .. . 0 0 0 .. . 0 0 0 0 0 0 mr ,i 0 1 0 .. . 0 0 0 .. . 0 0 0 0 0 0 mr^i 0 0 0 0 1 0 0 0 0 0 0 0 mr i 0 0 0 .. . 0 0 0 .. . 0 0 T2,l 0 0 0 m 0 0 .. . 0 0 0 .. . 0 0 0 0 0 0 -7*2,2 0 0 0 1 .. . 0 0 0 .. . 0 0 0 o o .ILL 0 0 0 0 0 0 £ 0 0 0 0 0 0 0 TO m T2,2 m n,2 TO TO o V o 2 TO m 0 2 nt 0 0 1 0 0 0 0 0 0 1 r ,i 0 o 0 0 0 0 0 0 0 ^-0 0 0 -r ,2 0 0 0 0 0 0 0 0 0 1 n TO n Thus the induced map is an isomorphism. We shall now deal with the case m = 0. Let a = (0, bi,.. • , bn, r , i , r 2 2 ) 2 , . . • , r i, r B> n>2 be fixed. We identify C with a {<s,t> x <p,q>eP 1 53 xP |pt = 0}. 1 ) 0 Chapter 2. Coordinate charts for Mo,o(P , 2) n Then the universal map / restricted to C is given by a f(<s,t> x <p,q>) = < sq,ps + b sq + qt, r ps + b sq + r qt,... , r ps + b sq + r , qt > . x 2A 2 2y2 nA n n 2 Consider the following P 1 V C a A, P 1 where the top copy of P has coordinates < s,t> and 1 f :<s,t>^ (< s , t > , < 0 , l > ) , x and the bottom copy of P has coordinates < p,q > and 1 f :<p,q>^ (< 1,0 >,<p,q>). 2 An element of H°(C , / * ( T p « ) ) is a global section of each of the sheaves / * / * T p " for a i = 1,2 such that these sections agree when p = t = 0. The universal map above restricted to the top copy of P is given by 1 / o f :< s, t >->•< s, b s +1, b s + r t,... , b s + r t > x x 2 2i2 n n>2 and restricted to the bottom copy is given by / ° J 2 -<P,q >->< q,P + hq,pr 2tl + b q,... ,pr 2 n>1 +bq >. n The open neighborhoods U and U cover /(C). The open set U pulls back to U via Xo Xl Xo s / o f and U via / o / , and U pulls back to Ui via / o / j where U = P - < 1, -b > 1 x q and U = P 2 1 2 Xl x - < - 6 i , l >. 54 x 2) Chapter 2. Coordinate charts for M (P , n 0fi The gluing data for Tpn (the matrix ^ following gluing matrix, M 0 given in the m ^ 0 case) pulls back to the on the top copy of P : 1 u (hs+t) s 2 0 2 (hs+t)(b s+r ,2t) 2 2 s (bis+t)(b s+r t) 3 0 0 3i2 bis+t s Mi = 0 0 0 0 0 0 (feis+f)(6 -is+r( _ t) s 0 0 (bis+t)(b s+r 2t) 0 0 n n 1)t2 2 V n n< bis+t s \ 0 bis+t s and to the matrix M where 2 / iP+biq) g 2 0 0 0 0 0 0 P+biq 9 o 0 2 (p+6ig)(r ,ip+6 g) 2 2 9 2 (p+fcig)(r ,ip+6 g) 3 3 n Mo (P+fclg)(r( -l),lP+ftn-lg) q 0 0 (p+6ig)(r ,ip+6ng) 0 0 B 2 V n P+biq q P+hq q on the bottom copy of P . 1 We shall deal with the top copy of P 1 first. A global section is given by an n-tuple of rational functions of the form \ 3 ' t=l,...,n j=l,2 where h = hs +1, and h = s, on Ui for j = 1 and U for j = 2. Where x 2 s Migi = g 2 Thus gi tl = - p i , and 2 = 0t,2v*i* + *) - 9i,iipiS 55 + % t) 2 Chapter 2. Coordinate charts for Mofi(P , 2) n for i — 2 , . . . , n. Hence the coordinate functions on U entirely determine the coordinate s functions on U\. The coordinate functions on U are not entirely free, in fact, if we let s 9i,j Q'itjS = b{jSt -\- Cijt then the above equation becomes s{a s -rbi st+c it ) 2 itl 2 tl it = (oj,2S +6i,2st+c 2i )(6iS+t)-(ai 2S +6 2 2 2 i) ) si+ci 2t )(6iS*+r 2i ). 2 1>2 > 2 ii Taking s = 0 we have 0 = Ci, - Ci 2 ) 2 r i ) 2 . i j 2 thus Ci,2 - ci i 2 r So gi 2 has three free coefficients and the #; have each two free coefficients (the s and 2 t )2 st coefficients) for i = 2 , . . . , n. So we have 3 + (n - 1)2 = 2n + 1 free coefficients in total, and the coordinate functions on U are of the form s / ai,2S +6l,2St+Cl,2t 2 \ \ 2 ? / Performing a similar analysis on the bottom copy of P we see that the global section is 1 completely determined by the section on U , and the section on U is of the form q / q a'i,2g + i,2Pg+ i,2P 2 6 c o' <? +6' P9+r ,ic' p 2 22 22 2 1 2 <,29 + n,2P9+ n.l 'l,2P 2 V 2 \ 2 6 r c These sections must agree when t = p = 0 therefore 2 = a' it2 for i = 1,... , n. Thus we have n + n + 1 + n + 1 = 3n + 2 free coefficients so F ° ( C , / * ( T p » ) ) = C a 56 3 n + 2 , with basis Chapter 2. Coordinate charts for jVfo,o(P > 2) n given on U by q / d \ dwx' d t d p 8 t d p d t d 2 p t d 2 dw ' 5 dwi q du>i" ' " ' s dw q dw ' s dw\' q dw\ 2 n n p d 2 d 2 \ ' s dw ' q dw J 2 2 n 2 n n Thus the section above, with respect to this basis is given by the vector: ( i,2> 2,2> • • • > n,2) &i,2> &1,2, ^2,2> &2,2, • • • , &n,2> &n,2, c ' a G a Now we will show that in the case m = 0 that # ° ( C , T ) = C a 1 ) 2 , Ci ) 3 n + 2 c i 2 also. We consider again the diagram P 1 V C C ^ a A P 1 where ii : 1^ -+ m, &i,... , &„, r , r , , • • • , r ,i, r ) C £/ 2]1 2 2 n ft)2 t 0»O>&i,--- A , r i , r , . . . , r i , r ^ C C / , 11 : 2 l 1 : V ^ ' ^ ~* A,r i,r 1 2 z :^ ! > ^ 2 2 > ->• We cover the top copy of P 2 2 f 2 , & , r ,i, r n 1 2 2 j n) n>2 , . . . ,r ,i,r„, ^ C f/ 2f2 n 2 9 , . . . , r , i , r , ^ C C/ . n n 2 p with U and U and the bottom copy with U and U . s t p q Abusing notation we shall also use U , U and U as a cover for the universal curve C as t p q before. We have that i2(P )QU UU . 1 q p The gluing data for T , </>^, where c <frt : fic|r/ 57 (Or/J " 3 t Chapter 2. Coordinate charts for A^o,o(P , 2) n <t>g : fic 0 , - > 3n (0 ) Ug is given by / -S 0 0 0 m f 0 0 0 0 I3n-2 0 0 and is given by 0 -$ 0 ; i 0 0 0 V0 0 Pulling back the gluing data back to the top copy of P , via i\ we have 1 \ -5 00 0 f 0 0 0 I311-2 V 0 0 / and pulling back via to the bottom copy of P via i we have 1 2 \ 0 -** 0 f 0 0 0 0 l3n-2 0 0 58 Chapter 2. Coordinate charts for M o(P , 2) n 0i Thus a section of i*(Tc) is given by / as +bst+ct \ 2 2 t 2 0 h k 2 \ hn-2 J on U and by t / as +bst+ct s 2 2 \ 2 0 ki k 2 \ on U . Similarly a section of i (Tc) s 2 h n J 2 is given by \ 0 a!q +b'qp+dp 2 2 Q 2 k[ k 2 \ Kn-2 59 J Chapter 2. Coordinate charts for A^ (P , 2) n 0]0 on U and by q ( a'q +b'qp+c'p P 2 \ 2 2 0 k[ key \ ^3n-2 / on U n Now these two sections must agree when p = t = 0 thus ' - * ] • ( 0 a' ki \ h -2 n o ^ = k[ y ^3n-2 / J Therefore we have the following 3n + 2 free coefficients: (b,C,b',c',ki,. Hence H°(C ,T ) a t d = C c t 2 d 3 n + 2 p « a ( j ) V a (J) ' q d . . ,fc n-2)3 in the case m = 0, and we have a basis d p (E)' q d 2 2 d (E) d d d d ' dV • • • ' 36 ' d r n 2 i l d ' dr , '''' ' 2 2 d dr y n dr , n 2 We now need to show that the matrix, giving the induced map from H°(C , / * ( T p » ) ) to a H°(C ,Tc) a with respect to our chosen bases, is injective. We shall calculate the induced map on each copy of P and glue. We shall use the neighborhood U of C . Now 1 q Chapter 2. Coordinate charts for Mofl(P , 2) n /-• P , t p t ( 1, - + 61 + - , r , i - + b + r , 2 - , • \ q so d(f) is given on U s XQ 2 2 q 2 r ,l s n P L + - 5 6 * « + n,2S r (with coordinates ^) by the matrix 1 72,2 d r ,2 n 1 7-2,1 '.5 dbi db ^,2 r ,i n 1 0 0 0 0 0 0 1 2 dbj 0 0 db 0 0 0 1 f 0 0 i 0 0 n dr ,i 0 dr ,2 0 dri,i 0 0 0 dr i j 2 0 0 0 dr ,i 0 0 0 0 0 2 2 n dr , n 2 \0 where tp = 0. The transpose of this matrix gives the map from / * ( T p « ) to T o In terms of the basis chosen the vector (b,c,b',c',ki,... ,fc n-2) 3 61 Chapter 2. Coordinate charts for .Mo,o(P , 2) n corresponds to a section g g 2 h k 2 \ hn-2 J on C C Ug where pt = 0. Under d(f) this maps to a b'Z + c'ZL + h - b l - 4 f r ,i {V* + d§) + k 2 r ^ + c'£) M r Bll (J) + fc + r , ( - & { - n+l 2 + ^-1+2(^-1) (f) (y; + d£) + A; 3 N - N (§) + 3 2 2 + h + r <)2 (-6f - ) + * c%) n 2 which corresponds to fc + r 2 fci + r (-6f-c£)+fc 2 l 2 n + 2 - c^) + A^ i i 2 +2(< (f) _ i ) (f) V fc + r , ( - 6 f - c f ) + A ; 3 n - 2 ( f ) n n 2 62 (}) + A W ^ D (f) k + r , ( - 6 J - c%) + n n + 2 / (f) / Chapter 2. Coordinate charts for Ma,o(P , 2) n when we restrict to the top copy of P , and look at the representative on U , and corre1 s sponds to V\ + c'$ * i + f k2 + r , (b^ + 2 l c'^)+k ^) n+1 (&'| + c ' £ ) + A:n-i 2 -i) ki + r + itl V N K +r B l l (tfj (i (f) + c ' £ ) + fc n- (j) 3 / 3 when we restrict to the bottom copy of P , and look at the representative on U . 1 q In terms of our chosen basis for H°(C .Tc), a (ki,... ,ki,... ,k , b', -b, b'r ,i+k i, n 2 ... n+ this corresponds to the following element of -br +k ,... 2t2 n+2 , b'r i + A; _ , - 6 r n> 3n 3 63 n > 2 , b'r i+k -i+2(i-i), it n + & _ , d, -c) 3n 2 -br +k (i-i), it2 n+2 Chapter 2. Coordinate charts for A^o,o(P , 2) n Therefore the matrix for this map is: 0 0 0 0 -r 0 2 j 2 0 -r<, 2 0 ^ -r„, r 2>1 0 0 0 0 0 0 ?v 0 0 0 0 0 hn-2 0 r„,i 0 2 0 0 0 j 0 64 0 \ Chapter 2. Coordinate charts for Mo,o(P , 2) n This matrix is also invertible with inverse M : m=0 'o 0 I 0 0 1 0 0 0 0 n 0 0 0 -r 0 -r W l 2 -r ,i 0 0 -r , n n 1 0 2(n-l) 0 2 P r o o f Consider the following diagram: [A -VZ ] 2 0 0 is eia/e 3 0 0 • Lemma 12 -1 2 l 2 0 -Ti,l 0 A AT ,o(P ,2) n 0 A "" 3 65 f Chapter 2. Coordinate charts for Mofi(P , 2) n The quotient map A ~ 3n Lemma 13 l -> [A ~ /Z ] 3n is etale and (j) is etale thus <j> is etale also. l 2 • (p is radiciel and representable. Proof We first show that the map <f> is radiciel. Thus we must show that the fiber over a point is equivalent to a scheme consisting of one point, i.e. is a groupoid consisting of a collection of isomorphic objects such that none of these objects has any automorphisms. Let X be a scheme and suppose a map X —>• A^o,o(P j 2) is given by the data n C A P f x n X We consider F(Spec(C)), where F is the fiber product, F -» X I 4 [Uo/Zv] M),o(P ,2) n In this case an element of the set X(Spec(C)) is a morphism from Spec(C) to X, so is given by a point x in X, and the image of this element in A1o,o(P , 2) is the pair (C , n x where C is the fiber over x. x A n object ofthe groupoid [U /Z ] (Spec(C)) is as follows: 0 2 4 B I Spec{C) 66 Uo fx), Chapter 2. Coordinate charts for .Mo,o(P , 2) n where B is a principal Z bundle over Spec(C) and g is Z -equivariant. If we take the 2 2 canonical section of B associated to the identity element Id of Z , then g(Id) is a point 2 in Uo, which we shall call p. We shall rename the bundle B and the map g in this p p case. If we let a be the non-identity element of Z then g (a) = ap. If p is invariant 2 p under the action of Z on Uo, then B has an automorphism, induced by the action of a 2 p on the fiber. If p is not invariant then its orbit contains a point p' distinct from p, and the bundle (B > —> p',g i) p (with the morphism g < induced by the action of Z on p') is p p 2 isomorphic to (B —>• p,g ), via the isomorphism which sends (Id,p) to (a,p') and (a,p) p p to (Id,p'). Given an object (B -+p,g ) e [U /Z }(Spec(C)), p p 0 2 this maps to the element of Afo,o(P , 2)(5pec(C)) given by ((B x n p Uo C ) / Z , / ) , which, 2 if we consider the section S of B associated to the identity element, is isomorphic to p (5 X r j C , / ) , which is in turn isomorphic to 0 If we assume that (C ,fx) (C , /). p G U(Spec(C)), then if (C ,fx) x x is a stack point there is one point p e UQ such that (C , / ) is isomorphic to (C , f ) by Lemma 8, but we have exactly p x x two choices for the isomorphism since (C , f) has exactly one non-trivial automorphism, p induced by the action of a ouC . Let us call these two isomorphisms tp : C -» C and p p x ip o a : C -» C . p x In this case F(Spec(C)) consists of the two objects (x,B ,ip), p (x,B ,ipoa). p We shall show that these objects are isomorphic and neither has any automorphisms. 67 Chapter 2. Coordinate charts for Mofi(P , 2) n We consider the automorphism of B induced by a. We need to find the induced morp phism from C = p x p Uo C to itself. Let (p, c) G C then (p, c) comes from the element p C where p = 7r(c). ((Id,p),c) in B x Uo Now (a(Id,p),c) = ((a,p),c) which lies in the same equivalence class as ((Id,p),ac) which corresponds to (p, ac) in C , thus the p induced map from C to C is a, and we have the following commutative diagram: p p (C ,p) A p (C ,f ) x ai x || (c , ) * F (c ,f ) p P x x so the two objects are isomorphic. This argument tells us also that neither point has any automorphisms, since any automorphism would have to be induced by the unique non-trivial automorphism of B , a. p If (C , fx) is not a stack point then there are points p and p' in U such that x 0 (C ,f), p (C ', f) and (C , fx) are isomorphic, and p and p' lie in the same equivalence class under p x the action of Z on U - Thus again F(Spec(C)) consists of two objects, in this case 2 0 (x, Bp, f^ o /), (x, B , f^ o f) 1 1 pl where B and B > are as above, (since fx is an isomorphism when restricted to C , and p p x / is an isomorphism when restricted to C or C' ). These objects are isomorphic, via the p p morphism (Id,p) -> (a,p'), (a,p) —> (Id,p'). This morphism induces a morphism given by a from (C , f) to (Cy, / ) , and we have the following commutative diagram: p (C ,f) f P ^ f x ai (C .,f) p (C ,f ) x || f ^ f (C ,f ) x x since f(a(c)) = f(c). So the two objects above are isomorphic. 68 Chapter 2. Coordinate charts for Ato,o(P > 2) n Neither of these objects has any automorphisms since neither B nor B > has any autop p morphisms. Therefore the map 4> is radiciel. To show that <f> is representable, we need to show that if X is a scheme then the fiber product F, F -> X i I [E/0/Z2] -> M),o(P ,2) n is a scheme. Let V be a scheme over C, we consider the diagram of groupoids F(V) ->• X(V) I I [t/ /Z ](V) 0 -> 2 A4o,o(P",2)(V) A n object of F(V) as a triple (h,B,*P) such that h : V — > X is a morphism, 5 is given by the data B U 0 I V and ip is an isomorphism from (V x x C , / ) , where S is the section of B C , fx) to (5 x associated to the identity element. A n automorphism of this object is an automorphism of B, r , such that the following diagram commutes: (Sx C,f) Uo A r l (Sx C,f) Uo (Vx CJx) x II A 69 (Vx C,f ) x x Chapter 2. Coordinate charts for Mo,o{P , 2) n If r is not the identity it must be nontrivial when restricted to some fiber of B, and both elements of this fiber must map to the same ( Z invariant) point p in Uo, thus we can 2 represent this fiber by (B -> v,g ), for v E V. Let x = h(v). p When we restrict the p diagram above to v, we have an automorphism of the object {x,B ,4>) of F(v), induced p by r. This is impossible, by the arguments above, thus r is the identity. Therefore t/> is representable as required. • We now consider the induced G action on Uo'm Lemma 14 The action of G on P induces an action on Uo which is given by the 71 m matrix X~ ( 0 2k 0 \ 0 \ Dx(A) 0 0 D (X) 0 2 j where D\(X) is the nxn diagonal matrix with d^i — X~ and D (X) is the 2(n—1) x2(n—1) kl 2 diagonal matrix with entries X~ , X~ , X~ , X~ , etc. to X~^ ~ ^ , A ^ k k 2k 2k n l k - - 1 ^ along the main diagonal. This action is compatible with the G action on the Afo,o(P )2) induced n m by the action N on P . It is also compatible with the Z action on UQ. 71 2 Proof Let x = (m, & i , . . . ,b , r , i , r n 2 2 > 2 ,... ,r i, r n i n>2 ) be a point in Uo- Then / restricted to C x is given by the rational map < s, t >-*< st, ms + bist +1 , mr ,\S 2 2 2 2 + b st + r 2 70 t ,... , mr ^s + b st + r , t > 2 2<2 2 n 2 n n 2 Chapter 2. Coordinate charts for A^o,o(P ) 2) n from P to P . We choose a different representative for (C , f) by composing / with the 1 n x projective change of coordinates, T\, given by the matrix \ 0 0 At Thus letting fx = / o r> we have / (< s,t >) = (st, ^s + bist + X t , ^ r 2 A ... k ,^ r Now we recall that the action of G N{\) = V 0 Xr t, k 2 k n 2 0 0 0 0 0 0 0 0 0 0 0 A 0 0 A 0 0 0 0 ... 0 0 0 0 ... a i Q 2 A""0 1 A " a Thus the point A.(C, / ) = (C, N(\of ) 0 2 2>2 2 n 0 where a = ^ and on = a -ki. + b st + 2 + b st + \ r , t } . 2 0 a o 2 A is given by the matrix m A s n A s 2 is parameterized x by the map / kn % fc(n-2) / m <s,t>-> (\Tst A~V*i^i) ^ R 2 ) L 5 2 k , t , \ 2 x r t ^j b2St + + , l^—s + bist + X t J t A o 2 ,X^ 2a k (jk-r s , 2 + b st + A r , t ) ) . 2 f c ntl n 2 n 2 kn Dividing through by A ~r we have < s, t >-> {st, X~ ms 2k 2 + X' hst + t , X~ mr s k A"**"* W „ , i s 1 2 3k + X~~ b st + 2 2k 2<1 2 + X~ b st kn n 2 + A~ f c ( n _ 1 ) r X~ r t ,... k 2 2>2 £ ) . 2 n ) 2 This corresponds to the map / restricted to the fiber Cx. where X.x is the point x (A" m, X'%, X~ %,X~ b , 2fc 2 nk n A- r f e , A" r , ,... , A f c 2 ) 1 71 2 2 f c ( n _ 1 V i, A**"" ^) 1 n j Chapter 2. Coordinate charts for A^o,o(P , 2) n in U. 0 Since G m acts with the same weight on and r^ for i = 2,... , n, we see that this 2 action is clearly compatible with the action of Z above. 2 2.5 • An Open Cover for M ,o(P , 2) n 0 We have so far constructed one affine open cell UQ isomorphic to A ™' with an etale map 3 1 <t> to M.ofl(J> , 2), and a Z action such that the induced map N 2 ^:[C/o/Z ]-^ATo,o(P ,2) n 2 is an open immersion. We have also calculated the induced G m action on UQ. What we would like is a method, using which, one can generate a collection {Ui} which taken together map surjectively to Mofl(P , 2). This is very easy to do. For our purposes n we only need two open cells, which cover the fixed locus under the A action. 1 L e m m a 15 Let P e PGL i be a projective change of coordinates for P . Let C and f 71 n+ be as before. Let be the morphism of stacks from A 3n+1 <f> P following data: C 7T -4 F 1 4 F 1 I Then fo-.A "3 1 ^AT ,o(P ,2) n 0 is etale, and the induced map from the quotient stack 4 :[A - /Z }-^M (P ,2) 3n P l n 2 Qt0 72 to Mo,o(P , 2) given by the n Chapter 2. Coordinate charts for Mo,o(P , 2) n is an open immersion. The image is the open substack Up defined by the following data: Up : (Schemes/C) —• (Groupoids) X -> U (X) P Where an element ofUp(X) is an element of Mofi(P ,2)(X) given by the data n {7r:C -+X,f :C ->P ), n x x x and such that this element satisfies the following: Consider f (H{) G T(C , f O(l)) x X where the Hi is the image of the coordinate hyperplane x Xi under P, and let Di = Z(f (Hi)) for i = 0,1 then x (i) The induced map TT : D —> X is unramified, 0 (ii) The divisors Do and D\ do not intersect. Proof This is an immediate consequence of Theorem 4, Lemma 11, and Theorem 5. • Corollary 1 Let X be a scheme and (v.Cx^XJx-.Cx^n, an element of Mo,o(P , 2)(X). For each x € X we can find an open subscheme x e U C n X and a projective change of coordinates for F , P, such that if C 1 v C x to U, then the data (n:C ^XJx:C -+P ), n u u gives an element of Up. 73 is the restriction of Chapter 2. Coordinate charts for M o ( P , 2) n 0 ] Proof Let x e X. We can find hyperplanes H and H such that fx(C ) D H D Hi is empty 0 (where C x x 0 is the fiber of Cx over x) and / x maps two distinct points of C x x into H . 0 Choose any projective change of coordinates P such that P maps the hyperplane X Q to # 0 and X = 0 to J?!. = 0 • x 2.6 Covering the Fixed Locus In this section we shall find two cells UQ and U\ which cover the fixed locus under the A 1 action given by M . We shall take UQ to correspond to our original construction, i.e. UQ = A 3 N ~ L with coordinates (ra, & i , . . . , r p), and the universal curve and map over UQ n are C and / . We shall identify which points on the fixed locus lie outside the image of UQ in A^o,o(P , 2), and find a cell U\ which covers the missing points. To do this we need n only find a suitable projective change of coordinates P (Corollary 1). Recall that the fixed locus consists of the following closed points: • Points (C, /) in M o ( P , 2) where f(C) is a conies in P n 0) I = (X — 2 c for ceA , 1 denoted by F M C 2X0X2 + c X | , X , X4,... 3 n given by the ideal , X) n . • The point (C, /) where C is isomorphic to two copies of P meeting at a point and 1 / maps each copy of P to the line given by the ideal 1 loo — (X2, X3,... and the crossing point maps to the unique A , 1 X ), n fixed point in P , namely the point n F =< 1,0,... , 0 >. This point corresponds to c = 00, and is denoted by F ^ . 0 74 Chapter 2. Coordinate charts for Mofl(P , 2) n The conic given by the ideal I meets V(X ) c 0 at two distinct points, <0,7,1,0,... ,0>, where 7 = - c , if c ^ 0 and at the point < 0,0,1,0,... , 0 > if c = 0. Thus F 2 M C is covered by UQ as long as c is non-zero. So we need to generate another cell, which contains the point F 0 M . There are many choices for this cell but we shall choose one which makes the gluing maps between the cells particularly nice. We notice that our conic does not intersect V(Xi,X — X ), and it intersects the hyper- Q 2 plane Xi — 0 at two distinct points. We can take H = X\ and H\ = X — X . 0 Consider the projective change of coordinates for P n 0 given by the matrix: I / n i \ 0 i1 1 \o o = \ 0 1 0 0 P 2 I) 0 In- 2 V P maps the hyperplane X Q 0 to HQ and the hyperplane X x will suffice. Lemma 16 Let 0 1 1 1 0 0 Vo o 0 75 \ \ 0 1/ In-S = 0 to H \ . Thus this P Chapter 2. Coordinate charts for !Mo,o(P > 2) n Let U\ — A " 3 - 1 with coordinates (m ,b ,... ,b' ,r' ,r , • • • x n 2X ,r' i,r' ). 22 n n2 Consider the rational map from U\ x P given by 1 f ((m'b[,... ,r' ),<s,t>) = P nt2 < rn\r' +l)s +(b[+b' )st+(r' +l)t , st,m'r s +b' st+r' t ,... 2 2 2l 2 2 2>2 ,m'r' s +b st+r t 2 2l 2 2 2>2 nA 2 n n2 > Let C be the blowup of Ui x P along the subvariety where t = m' = 0, as before and let 1 fp be the induced morphism from C to P" . Then the induced map from Ui to jMo,o(P , 2) 1 n is etale and the image contains the closed point F . M 0 Proof Follows immediately. • 2.6.1 Transition Functions Since each conic in P is contained in a plane, we calculate the transition functions by n equating planes. First however we observe that the projection map p: P -> n < x ,xi,... ,x 0 n > P 2 <a; ,a;i,X2> -» 0 induces a map Ui, -> n (m, , bn, r i , r for i = 0,1, where U 2 ) 2 j 2 , . . . , r ,i, r n is the is the i th itk n>2 ) U i>2 (m, b b , r , i , r u 2 2 2>2 ). member of the cover for M o(P ,2). k 0t Uoi k be the intersection C/,fe and t/i^, we have a commutative diagram: t 0 Ui, Uoi 4 4" n Ul, 2 >n <r- UQI,2 76 —> Uo, n 4' —> Uo >2 If we let Chapter 2. Coordinate charts for Mofi(P , 2) n where all the horizontal maps are two to one. We shall initially calculate the gluing maps in the dimension two case and then calculate them in the general case. One can readily check that the map < s, t >—st, ms + bist +1 , mr ,iS 2 2 + b st + r t > 2 2 2 2 in the case r ,\ i 1 2 r 2}2 2 , 2 parameterizes a conic (which is degenerate when m = 0) with equation ((b - r , i 6 i ) X + r , i X i - X ) ((b - r h)X 2 2 0 2 2 2 2y2 + r , * i - X ) + m(r ,i - r , ) X 2 0 2 2 2 2 2 2 2 = 0 and the map < s,t > - K m'(r in the case r' 21 + l)s + (b + b[)st + (r + l)t ,st,m'r s 2 2>1 2 2 2>2 + b' st + r' t > 2 2 21 2 22 ^ r' parameterizes a conic (which is degenerate when m! = 0) with 22 equation ((6' - r'^b'JX, + r' (X - X ) - X ) {(b' - r « )X + r' (X - X ) - X ) + f 2 2A 0 2 2 2 vl m'(r' - r' ) X 2 2tl 2 2>2 l l 2>2 0 2 2 = 0. Since conies corresponding to points in (7o, have the property that they do not pass 2 through the point < 0,0,1 > (the intersection point of the hyper planes X — 0 and 0 Xi = 0) and the line X = 0 is not tangent, if a conic in Ui lies in (7oi,2 then it has the 0 t2 following properties: «2 = W,l + 1 ) ( ^ + 1)#0 and Ki - r' ) {(b[ + b' ) - 4m'(r 2 2 2<2 2 => u = (b[ + b' ) - 4ro'(r 2 x 2 77 2>1 2>1 + l)(r , + 1)) ^ 0 + l)(r 2 2>2 2 + 1) # 0. Chapter 2. Coordinate charts for Mo,o(P , 2) n Therefore on the intersection we can invert u\ and u . Similarly in Uo, coordinates conies 2 2 corresponding to points in the intersection do not pass through the point < 1,0,1 > (the intersection point of the hyperplanes X\ = 0 and X — X = 0), and the line X\ = 0 is Q 2 not tangent so we can invert v = (62 ~ ^2,161 - 1)(&2 - r ,2h 2 2 - 1) + m(r ,i - r 2 2 ) 2 ) 2 and =»• vi = b\ - Am ^ 0. Equating the coefficients of the conies above we have the following relations: m = , 1 2,l) 2,2 r r mv 2 v\ _ 26 - (r ,i + r )&i - 2(6 - r ,ifti)(fr - r 6i) - 2ro(r ,i - r , ) ± (r ,i - r — 2v -5i&2 + 61 + 2ro(r i + r ) 2 2 2 2|2 2 2 2 2>2 2 2 2 2 2 2> 2|2 bib ) - 2m(r ,i + r ) 2 2 2|2 «i Going in the opposite direction we have m = mu 2 (62 ~ 6i)(r ^2,1, ?"2,2 2|1 +r + 2% - 2 ^ ^ ) ± (r' - r' )^ui 2>2 2A 2>2 2u 2 b[ + b' 2 Ui b = 2 b' (b'i + b' ) - 2m'(2r' ^ 2 2 2 2 +r +r 2 < 1 2>2 ) «i These relations hold also in the case where r ,\ = T 2 2 ) 2 and r' 2l = r' . Therefore we have 22 the following maps of rings, in Ui coordinates: j2 T(u , ) 0 2 C[m,6i,6 ,r 2 2>1 ,r ] 2)2 -> r([/ ) 01>2 -> C m', 6' 6 , r 1 1? 2 ^ 78 r , W , —, — ^ Ui u 7 2 v A i 2 2 2 I ({W'f - ui) 2 | 2 )0^ Chapter 2. Coordinate charts for M ,o(P , 2) n 0 where m m'u 2 - » —5- ftj +w&2 }J 61 , 0 2 1 _ + ~ 2m'(2r r 2]1 + r 2)2 2 | 1 + r 2>2 ) —r Ml (b'2 - t j ) ^ , ! + r' , + 26 - 2 6 ^ , , ) + (r 2 2 2 2>1 - r ^ W (6 - b[)(r' + r , + 2b> - 2 6 ' ^ , ^ ) - ( r ^ 2A 2 r 2 2 2 2 2 ri^W' -> 2u 2 In particular we see that the points (m'X,b' ,r' ,r' ,W') 2 2tl 2>2 and (m'^b^r'^r'^-W') in t/01,2 both map to the same point in Uo, and 2 {m\b\,b' ,r' ^ ,W') 2 2 2 and (m',6' ,6' ,r ,r 1 2 2il 2)2 ,-W) map to the same point in (7 . Thus we have four points in 1)2 C/01,2 giving us the same stable map. In order to calculate the transition functions in the general case we observe that if points (m, 61,... , b , r , i , r n 2 2 ) 2 , • • • , r ,i, r , ) n n 2 and ( m ' , . . . ,b' ,r ,r ,... , r n 2 1 79 2 2 n l ,r n 2 ), Chapter 2. Coordinate charts for A4o,o(P , 2) n with r i ^ r 2 and r'^ ^ r'^ for some 2 < i, j < n, represent the same conic in P then n it 2 it the plane spanned by the points < 1, bi, b ,... , b >, < 0,1, r i , . • • , r„,i >, < 0,1, r , , . . . , r 2 n 2 ) 2 2 n<2 > must be the same as the plane spanned by the points <b' + b[, 1, b' ,... , b' >, < r' 2 2 n 2jl + 1,0, r 2 j l ) . . . , r' >, < r njl 2>2 + 1,0, r 2 ) 2 , . . . , rj, >2 >. Therefore the latter three points can be written as a linear combination of the former three, and vice versa. In this way one can solve for the rest of the coordinates and we get the following relations: r "t,l ± Vj,2y/Vl i , D i,2 r 2v -b bj + 2m(r + r ) 2 x b' iA <>2 where -2m(r Vi,l - r 2)1 2)2 +bibi(r + r 2>l Vi, 2 = -6i(ri, r 2 2|1 )(r 2)2 - r ) - b\(r r i)1 ii2 ) + b b (r x 2 itl i>2 2 2 > 2 r ) M + r ) - &i(r i + r ) - 26 6j + 2b i)2 - r , r ) + 6j(r i - r 2 + r 2A M 2i if 2)2 ) - 6 (r 2 i>2 ijl 2 { - r ) + (r i>2 M - r ), i>2 and ^,1,^,2 2u 2 -b\% + &;) + 2m'(r^ + rj, + r[^ 2 2 + r^r a>1 ) where = "(&2 + +2^ + ^ ( r *,2 W = + T - - + r^r^a + r j , r ) 4- ( 2 r r 2 2 i l +r 2 j 2 2 ) <1 - <2 + < 1 < 2 - < 2 < 1 for i = 3 , . . . , n. 80 2il 2il 2)2 +r 2 ) 1 + r )&; 2)2 Chapter 2. Coordinate charts for Mo,o(P , 2) n 2.6.2 T h e F i x e d Locus We previously described the fixed locus in terms of geometric points of Mofi(P , 2). We n would now like to express it in terms of the coordinates of the two neighborhoods. The fixed locus is contained in the subvariety V(X , X±,... , X ) of P , thus we can project n 3 n the fixed locus into P . If a stable map corresponding to a point in Uo or Ui has its image 2 contained in V(X$,X±,... , X ) we must have, in each neighborhood respectively, n bi = K = r i = r it i%2 = r- = r- = 0, for % = 3 , . . . , n. ;1 >2 Thus when dealing with fixed conies we identify each with a 5-tuple of complex numbers. In P , each conic is given by a polynomial as above. Thus solving a series of equations, 2 the fixed conic (for c ^ 0) 1 2 —X, c 2 c XoX + X 2 2 7 corresponds to the two points The stack point (corresponding to c = 00) corresponds to the origin. If F(XQ, Xi, X2) is a conic corresponding to a point in Ui then F(1,0,1) = 1 thus we express the fixed conies in the following way for c 7^ 2. Solving a system of equations we see that this conic corresponds to the two points ( ^ T ' °' °' 7^2' °) ' °' °' °' rh) 81 = (m '' ' ' ' &; 625r2 1 r2,2) Chapter 2. Coordinate charts for Mo,o(P , 2) n 2.7 A vector field on M , ( P , 2) n 0 0 In the last section we gave a method using which on can construct a family of affine varieties {Ui} such that each U{ is isomorphic to A " , and in each case we have an 3 ieI - 1 etale map from Ui to Mo,o(P ,2). In this section we shall find the vector fields on UQ n and U\ which are pull-backs of the vector field on .Mo,o(P > 2). n Lemma 17 Let UQ be as in Section 2.4. Then the vectorfieldon ' m ( - 4 6 i + r ,i + r , ) 2 2 UQ is ^ 2 - 4 m - b\ + b 2 -2m(r ,i + r ) - b b + h 2 2j2 x 2 -2m(r i + r ) - b b + b i ij i)2 x -2m(r„,i + r B|2 { i+ ) - hb n r ,i6i - r , i - b + r , i 2 2 2 r , &i - r\ -b 2 2 2 3 +r 2 3 j 2 >"i,i&i - i,i 2,i -bi + r( i),i r r i+ n,2^i - r r ,2 -bi + r ( i ) it2 2 i+ i2 r ,i&i - r„,ir i - 6„ n \ 2> fn,2h - r , r , - 6 n 2 2 2 / B with respect to the basis ( d_ d_ d_ _d d_ Vdm"' db[''"" ' db ' 9 r , i ' d r n 2 _d 2 > 2 '" ' ' 9r d_\ n > 1 ' dr , ) n 2 Proof Let V be the section of Tp» given by the A action M . Let <f> be the map of stacks given 1 82 Chapter 2. Coordinate charts for M o(P , 2) n 0t by the data C A P 7T 4- as before. Since the map of stacks given by this data is etale, we have an isomorphism (C ,f*T ») H\C ,T )^H* a p Suppose we restrict to the locus of points a e UQ such induced by / , for each a e A ~ . 3n a c l ) G UQ with m / 0 ) . that C is irreducible (i.e. a = (m,bi,... ,r a n%2 this isomorphism is given by the matrix M ^ m 0 Then the inverse of of Lemma 11. We shall find the pull-back of the section V using M ^ o - We shall restrict V to the m neighborhood U of P . n X O Recall that the A action given by M is as follows: 1 < XQ + tXi + . . . + —.X , n t. < XQ, X \ , Xi + tX X N is Thus the induced vector field on U XQX 2 — x\ 2 XQ XQXJ, — ' 2 + . . . ,X N . . . X X\ + XQX 2 n 2 XQ 83 > — X —\X\ n X X\ n Chapter 2. Coordinate charts for M ,o(P , 2) n 0 Pulling back via / to C , we have a section of / * 7 p « given on f~ {U ) c U by: l XQ ( t st(mr2,is +b2St+r ,2t )-(ms +bist+t ) 2 2 2 2 \ 2 2 st st(mr3,is +b3st+r3,2t )-(mr2,is +b2St+r ,2t )(ms +bist+t ) st 2 2 2 2 2 2 2 2 2 2 2 st(mr i, s +bi ist+ri i t )-(mri,is +bist+r 2t )(ms +bist+t ) 2 i+ 2 1 + + I \ 2 2 tl 2 2 ii - ( m r , 1 s +b 2 n st+r , 1 ) (ms +b sW 2 n n t+t ) 2 2 J 2 3 / which, written in increasing powers of t, is / (-m )s +(m(-2fci+r ,i))a (+(-2m-fe +6 )s i +(-2fc +r2,2)^ -t st 2 4 3 2 2 2 2 3 2 2 \ 4 1 2 {-m r2^)s +(m(-b2-biT2 i+rz i))s t+{-m{r2,\+r2a)-bib2+b3)s t +(-^^ 2 4 3 y 2 2 l ^2 (-m r 2 )s +(m(-6 -cnr , +r 4 ( i + 1 ) i l i i 1 ( i + 1 ) , ))s^ 1 ^2 •• (-m r )s +(m(-fe -r ,i&i))^ t-|-(-m(T- ,i+r ,2)-6nfei)a t +(-&n-fcir ,2)st +(-rn,2)t st 2 4 na 3 n n 2 T1 n 2 84 2 2 3 n 4 / ) Chapter 2. Coordinate charts for Mo,o(P , 2) n This corresponds to the following element of C with respect to our chosen basis: 3 n + 2 \ m(-26i +r ,i) 2 -2m - b\ + b 2 -2h + r , 2 2 -1 m(-bi - 61^,1 + r( i),i) i+ -m(r i + r ) - M i + b i) i)2 i+i - h - 6ir + r(i+i), i)2 2 m ( - 6 „ - r ,i&i) B -»n(r i + r ) - 6„6i ni n>2 - 6 „ - &ir„ 85 )2 Chapter 2. Coordinate charts for Mo,o(P , 2) n Multiplying by M ^ m 0 we have —m 2&i - r ,2 2 -1 m(-46i + r , i + r , ) 2 2 2 - 4 m - b\ + b 2 -2m(r ,i + r , ) - M 2 + b 2 2 2 3 -2m(r« 4- r ) - M i + ]1 i)2 - 2 m ( r „ , i + r ) - 6i6 n>2 r-2,i&i - r | p l - 6 + r ,i ?*2,2&i - r | i 2 - &2 + r _ r 3 ) 2 r i>2 J>,1&1 _ 2y2 r -bi + r ( i i + ) ) 2 n,1^2,l - &n r ,2&i - r B 3 i , l 2 , l - 6i + »"(»+i),i r^&i - r r ^ 2 n B | 2 r 86 2 ) 2 - b n y Chapter 2. Coordinate charts for Mofi(P , 2) n When we push forward to UQ we forget the first three coordinates. Thus our vector field on UQ — V(m) is given by ' m(-46i + r , i + r 2 ) 2|2 ^ -Am -b\ + b 2 -2m(r ,i + r 2 2i2 ) - &i& + 63 2 -2m(ri,i + r ) - bib + b i|2 t i+1 -2m(r i + r„ ) - 6 6 n> |2 x r ,i&i - r\\ -b n +r 3 ) 1 r 6 i - r\ - 6 + r 3 > 2 2 2 2(2 2 2 - n,ir ,i - 6» + 2 ^,261 - r r iy2 -bi + r %2 ( i + 1 ) i 2 r ,ibi - r„,ir i - b n \ 2) r bi - r , r n>2 n 2 2 i 2 n - b J n This extends uniquely to give a section of Tu - • 0 We now need to calculate the vector field on Ui. Recall that when tVi is identified with A 3 n _ 1 with coordinates (m',b' ... v ,b' ,r' ,r' ,...,r' ,r' ) n 87 2A 2i2 n>1 nt2 Chapter 2. Coordinate charts for A^ ,o(P , 2) n 0 then the universal curve for U is the blowup U\ x P and the universal map f 1 x P is P o f where / is the universal map for U and 0 / / n0 i1 1 1 \ 0 1 0 0 P = \ 0 1 ) \0 0 V L* n - 2 7 Therefore in order to find the vector field induced on U\ we pull back the element of Tp"(J7 ) given by the vector field V via P giving an element of 7 p » ( [ / ) which using Xl X o the matrix M ^ (with all the entries replaced with the corresponding coordinates for m 0 Ui) above gives us a vector field on Ui. We choose affine coordinates for U which pull back via P to give the standard affine Xl coordinates for U , i.e. we choose the affine coordinates Xo Xp - X 2 ) Xi X2 £ 3 %n > )•••) Xi Xi Xi In these coordinates, the vector field V on U Xl 'Xi(xi - X ) - X (x 3 2 - X) 0 XiX 2 3 ' ™.2 is given by the following: - x\ 2 XiX - X-X n n '" * " ' X ~X X 2 2 n ' 2 Pulling back via P we have the following /x (x - x ) 0 0 3 ~2 XQ XX X XQXS 2 ' - x\ 2 '''' ' XQ XX 0 n n 1 XQ 88 -x x - X -iX 2 2 ' 2 XQ n 2 Chapter 2. Coordinate charts for Mo,o{P , 2) n We pull back via / and list in increasing powers of t as before -(m') r' f 2 2>l —m'r' — m'b' — m'b\r' 31 2l 2 l-b' -b[b' -m'(r' +ry 2 3 2!l ~ 3,2 ~b' — b\r r 22 2 b' - {b' f - 2mr' r' 3 2 r 3,2 2tl 2&2 2,2 — r ~ ' 2 'i,l m b ~ r K+i ~ 'i 2 - m'r' r' b 2t2 r'i+1,2 ~ 2<,2 ~ b b ~ ™' 'n-lA,2 r 'n,2 ~ 2 'n-l,2 r b m b ~ 2 n ~ 2 n,2 b> T ' n-l,2 2,l r r - 89 r ' 'n 2,l m b -b' b' - m'r' ^' n m b ~ r 'A,2 b ~~ 'n-X 2,2 r - ' 2 'n,l r - m'r'^r'^ b iA ' 'i 2,l m b b - 'n-l 2 2<2 - m'r r 2t2 n>2 'n 2,2 b r r 2)1 Chapter 2. Coordinate charts for A^o,o(P , 2) n We then multiply by M ( and push-forward to Ui to get the following vector field: m # 0 • -m'(r' + r 3A -2m'(r 2il + (r 3>2 + r'^V, + 2b' ) 2>1 ^ 2 + r y + l - b' - b[b' 3 -4m'r r 2)1 2 - [b' f + b' 2)2 2 3 -2m'(rJ ^ +rJ ^ )-6 6J + 6{ i2 -2m'(r _ r n li2 2>1 |1 fl i2 +r _ r n -2m\r' r' nt2 M a 2 i 2 +1 ) - b'^ n>1 2t2 2 2!l r l ( 2,2) ^i 2 r 2 — 2il 3tl &2 2,2 + 3,2 r r r'i,A,i + r'iAA ~ 'Ax b r r r n + r' 2 3>l 2,2 3,2 + n + r' r' )-b' b' 2!l r'2,ir' + (r' ) b' -b' r' r + b' + * i + M i,2 3,2 + i,2 2,2^'l ~~ 'i 2,2 + 'i+l,2 r r b r T V n - l , l 3 , l + n - l , l 2 , l ^ i ~~ &n-l 2,l + 'n,l r r r n-l,2 3,2 + 'n-l,2 2,2 '\ r r r r r ~ K-l 2,2 b r> + 'n,2 r r'nA^+r'^r'^-by^ \ 'n,2 3,2 + 'n,2 2,2 'l r r r r b — J ^n 2,2 r in terms of the basis / d _d_ _d_ _d d_ 90 _d d_\ Chapter 3 The Cohomology o f ¥ , ( P , 2 ) n 0 0 3.1 The Hypercohomology of M ,o(P , 2) n 0 Let {Ui} be the collection of affine varieties each isomorphic to A 3 n _ 1 constructed in Chapter 2. Our aim is to calculate the hypercohomology H°(.Mo,o(P ,2),/C*) where K? = Q,~ , n p for p < 0, is the complex of sheaves associated to the vector field, with differential i(V). We can achieve this by calculating the hypercohomology of the following double complex, using our etale cover {Ui}: t ••• -»• t ®i<j<kto {u ) 2 ijk -»• ®i n(Ui ) <j<k ••• ©i<jfi (yij) 2 t -»• 3.1.1 -> e^ ^) 2 ®MUi) Jk ®i<jO(Uij) t -+ ®i<j<kO(Ui ) -> o -> 0 t 8i<jfi(^ij) t ••• -> Jk t t (*) t -> ©<o(c/i) -> o P r e l i m i n a r y Ideas In order to calculate the hypercohomology of the complex (*) above we shall use the concept of quasi-isomorphic complexes. 91 Chapter 3. The Cohomology of M (P , 2) n 0 ) 0 Definition 10 A map j : C* -)• IC between complexes of sheaves is a quasi-isomorphism if it induces an isomorphism on cohomology sheaves: j * : rl (C) -> H {)C*) q>0. q q In our case KP = Q,~ for p < 0, and the differential i(V) is a map p i(V) : KP '-»• KP~ l and "H is the cohomology sheaf associated to the presheaf p w ( n ) < t 7 ) " • L e m m a 18 Suppose the map of complexes of sheaves on X, j : C ^ K* is a quasi-isomorphism. Then the induced map on hypercohomology j, :IF(X,£*) ->W(X,1C) is an isomorphism. Proof One compares the spectral sequences associated to a each double complex. • Thus our idea shall be to replace the complex of sheaves K* with a quasi isomorphic complex whose hypercohomology is more accessible. ways: 92 We shall simplify in four main Chapter 3. The Cohomology of M ,o (P , 2) n 0 (i) We shall show that one need only consider the members of the open cover U which meet the locus where the vector field vanishes. Thus since U and U\ cover the zero Q locus (in the notation of Chapter 2) we need only consider these members of the cover. (ii) The vector field is given, on each of U and U\, by a collection of 3n — 1 regular 0 functions. Suppose that a we have subsequence of length (3n — 1) — 1 in each case which is a regular sequence (Definition 12). Then the cohomology sheaves rl~ p vanish for p > 2 so we can truncate the complex of sheaves in question to 2 terms. (iii) We shall introduce the idea of a companion pair of vector fields for the vector field V (Definition 11) to simplify the remaining modules in the double complex. (iv) After simplifying as above, we reduce our problem to calculating the hypercohomology of a four term double complex. We observe at this stage that the group Z x Z acts on each of the four terms and all the maps are equivariant with respect 2 2 to this action. Thus the complex has a Z x Z invariant subcomplex and taking 2 2 the Z x Z invariant subring of the hypercohomology is the same as taking the 2 2 hypercohomology of the Z x Z invariant subcomplex. 2 Definition 11 A 2 pair of vectorfields(WQ,WI) on UQ and U\ respectively is called a companion pair for the vectorfieldV it has the following properties: (i) Wi does not vanish on Ui. (ii) The kernel ofWi, K-i, is mapped injectively by i(V) to Ou for i = 0,1. if (Hi) W = fW 0 x on UQI, where f e O* {U I). UQ 0 As a result of the three statements above we shall show that the double complex (*) is quasi-isomorphic to the four term double complex given in the statement of the following lemma: 93 Chapter 3. The Cohomology of M (P , 2) n 0 ) 0 Lemma 19 Let V and U be as above. Suppose that the zero locus of the vector field V is covered by U and U . Let V be given by the collection of rational functions 0 x Hi — (vi,i,--. ,Vi k) on Ui, and suppose that for i — 0,1 the sequence v has a regular { t subsequence of length k — 1. Suppose also that (W , W\) is a companion pair for V. Then 0 the double complex (*) is quasi-isomorphic to the following four term double complex: o n /(i(v)n 0 0 t t + Ko)(u i) 2 Uo Uo -»• 0 t 0 o /(i'v)Ko){u ) Uo o 01 t (B^itouJiiW^+lQiUi) -4 ®i^,iOuJ{i{y)lCi){Ui) 0 where Ki is the kernel ofWi. Proof Lemma 20 below will show that we can throw out all the Ui except UQ and U\ since we can assume that the others do not meet the fixed locus (by replacing them with smaller affine open sets if necessary). Lemma 23 will show that the cohomology sheaves vanish for p > 2. Lemma 24 will give us the four terms of the double complex. The conclusion of Lemma 19 is then immediate. • Lemma 20 Let p>0 and suppose x lies in one of our affine open covering cells U for Mofl(P ,2), such that the induced vectorfieldV does not vanish on U. Then the stalk n 'H~ is trivial. p To prove Lemma 20 we shall use the Koszul complex. But first two definitions. Definition 12 Let R be a ring and M and R-module. A sequence v\,V2,.-- called a regular sequence for M, if the following conditions hold: 94 ,v € R is r Chapter 3. The Cohomology of M ,o (P , 2) n 0 (i) V\ is not a zero divisor in M, and for all i = 2,... ,r, V{ is not a zero divisor in M/(v ,v ,... 1 ,Vi-i)M. 2 (ii) M/(v ...,v )M^O. lt r In particular we shall be interested in the case where M = R — C[xi,... , In this case . . . , v is a regular sequence for R , then it is a regular sequence for the localization if r RM for each maximal ideal M such that M ~D (i>i,... , v ). r D e f i n i t i o n 13 Given a ring R and a sequence vi,... , v in R. We can define a complex r K as follows: K = R, K = 0 for p ^ 1,... , r and p 0 p ~ •^ *i.-.*p e *i<ii<..-<ip otherwise, where ij = 1,... , r and e ,.^ iu ip is a formal basis element. Thus K is a free p R-module of rank ( ). The differential d : K —> K _\ is defined by n p v p de ..., = iu ^- ) ~ i iu...,r ,..,ip 1 ip i lv e i forp > 1, and for p = 1, dej = Vj for j = 1,... , r. This differential has the property that d = 0 and the resulting complex (K , d) is called the Koszul complex, and is written 2 p K.{v ... v ) = K.{v). u t r Suppose M is an .R-module then K.(v,M) is defined to be K.(v) ®R M. If C* is a complex of J?-modules then C.(v) is defined to be C* <B> K.(v). It is easy to see that K.(v u ... ,v ) = K.'vi) <8> K.{v ) ® • • • ® K.(v ). r 2 r L e m m a 21 Let C* be a complex of R-modules, and v 6 R. Then we obtain an exact sequence of complexes o-»c\, -•£.(«) ^ c : ->0, 95 Chapter 3. The Cohomology of M ,o (P , 2) n 0 where CI is the complex obtained by shifting the degrees in C* up by 1 (i.e. C' = C -\). p v The associated long exact sequence in homology is ••• H (C.) -+ P H^iC*) H (C.(v)) p H -i(C*) ( T 1 —>•••. p Proof: See [13, Theorem 16.4, p. 127]. Lemma 22 Suppose that ... , v ) is a regular sequence in R, then r H (v, M) = 0 for p> 0 and H (v, M) = M/vM. p 0 Proof: See [13, Theorem 16.5, p. 128.]. • Proof of Lemma 20: Since U is an open subset of A k = Spec(C[xi,... for some k, the vector field is ,Xk\), given by a collection of regular functions vi,... ,v k G T(i7). Let x G U then since the vector field does not vanish at x we can assume that at least one of the Vi is a unit in Ou, , x let us assume that Vi is a unit in Ou . tX coordinates xi,... ,x k Let Ou, = R, and suppose that A has k x then the complex associated to the vector field restricted to the stalk at x: ...-•fi^ns- -*... 1 can be identified with the Koszul complex by identifying e^...,^ with dxiAdx A.. 2 Since v\ is a unit, HQ(K.(VI, . . . ,Vi)) .Adx . k = 0 for i = 1,... , r , because if we take / G 96 Chapter 3. The Cohomology of M , (P , 2) n 0 0 Thus Hi = 0. We shall show the other homology groups vanish using induction on i. Clearly H (K.(vi)) = 0 for all p. If we assume that H \K.(v p p u ... , vi)) = 0 for all p > 0 then the exact sequence • H (K.(v p ... , v^) -» H (K.(v u p ... , v i)) -> Hp-iiK.ivu ... , vi)) ->• • • • u i+ shows that H (K.(v ... p Thus H (K.(vi,... p H (K.(vi,... p ,v )) = 0 for p>l. u i+1 , v i)) = 0 for i = 0,... , r—1 andp > 0, so in particular H (K.(v)) — i+ p , v )) = 0 for all p > 0. Now via our identification H~ = H (v, R) so p r p ri- = 0, for p > 0 , p as required. • L e m m a 23 Let U be an affine open subset of A and let the vectorfieldV be given by k (vi,... , v ) on U. Suppose (v\,... , v -i) is a regular sequence. Then for all p > 2, the k k cohomology sheaf %~ = 0. p Proof: Let x e U. We shall show that U~ = 0, when p > 2. We assume that the vector field p vanishes on U since if it did not we would be done by Lemma 20. Let x £ U, and let R =0 as before. By Lemma 21 above we have a long exact sequence Ux ...-». H (K.(vi,... p ,v )) , v ^)) = H {K.(v v ^)) = 0 for p > 2, H (K.(vi,v )) p Since H ^{K.{v ... p ,«*_!)) -> H (K.(vi,... u k p u k k 0 for p > 2. Thus U = 0 for p > 2 as required. • p x 97 # _i(#>i> • • • >«*-i)) { p p k ~ ^ 1 V k - Chapter 3. The Cohomology of M ( P , 2) n 0 > 0 We would like to have a simple way of deciding when a pair (Wo, W\) of vector fields is a companion pair. The most difficult part ofthe definition to check is part (ii), i.e. that the kernel of a vector field W on Ui is mapped injectively by i(V) to Ou - The following t lemma gives a condition on the vector field V which ensures that this property holds, and gives us an explicit description of the four terms of the double complex in Lemma 19. Lemma 24 Suppose that A has coordinates k on U C A . Suppose i(V)dyi k = V{ ,y yi,... k and W is the vector field for i = 1,... , k — 1 where (v\,... , v k is a regular -i) sequence in T(U). Then i(V) maps the subsheafK offlu generated by < dyi,... , dy -i k injectively to OJJ (this subsheaf is the kernel ofW) and tlu/iUVn + K)^ (T(U)/(v v -!)) dy , u Ovl{i(V)K) ^r(U)/(v ... u k k ,v . ). k 1 Proof We shall show that i(V) maps the stalk K injectively to Ou, for each p e U. First v P suppose p G V(vi,... , ffc-i), then (v\,... , v -i) is a regular sequence in Ou, - The stalk k P K for p € U is generated by dyi, for i = 1,... , k - 1 as an Ou, module. Consider the p P submodule of Vlu, generated by dyi. p Then i(V) clearly maps this submodule to Ou, P injectively since Ou, is an integral domain. Consider the induced map i(V) , v UJ/) : ^ /(i(V)n + Ou,pdyi) 2 1 p UtP Ou, /viO p t i(V): t n ji(y)(^ ) v UtP -> p 0 UtP The submodule i(V)Sl\j contains in elements of the form i(V)(dyiAdyj) = for j = 2 , . . . , k. Thus i(y)ti{j p v dyj-Vjdyi x + Ou, dy\ contains elements of the form v dyj for P x 98 > Chapter 3. The Cohomology of M ,o(P , 2) n 0 j = 2 , . . . , k, so W ( » G 0 f # , + Ou, d ) - ( 0 P (r(tOp/«i) Vl ) /i(tOn* „. \j=2,...,A: / Suppose next we consider the submodule of 2 generated by dy , and the induced map i(V) from this submodule to Ou, /viOu, , 2 Wf • ^u, /(i(V)^ P + Ou, d p p + O dy ) yi u>p 2 -> UtP 2 2 UtP u>p +o d ) u<p Then this induced map is injective if and only if fv 2 p p t n /(i{v)n / = 0 in Ou, /viOu, , p 0 /{vi,v )Ou, t Wf • p OujviOu,,, yi = 0 in Ou, /viOu, p p implies that this is exactly the condition that v ,v is a regular sequence in P x 2 Ou, - Suppose that this is the case, then in the same way as before we observe that since p i(y)Vtij contains elements of the form i(V)(dy A dyj) — v dyj — Vjdy for j = 3 , . . . , k, i(V)Q + Ou, dyi + Ou, dy contains elements of the form v dyj for j as above. Thus p 2 /p 2 p p Qu, /(i(V)nl p p 2 2 2 + Ou, d p yi 2 + Ou, dy ) p 2 <* ( 0 (r(£Op/fa,t*))dVi) \j=3,... ,k J Thus continuing in this way we see that if the sequence vi,vs,... ,v -i is a regular k sequence in Ou, for each p then K is mapped injectively by i(V) to Ou, and in particular p p P we have the following: SluJ(i(V)n 2 Ujt +K ) * p {r(U) /{v dy , p u k and OWW W(«i s In the case where p V(vi,... sub-sheaf of fir/,p/i(V)fi 2 /jP , v -i) we can assume that v invertible in Ou, thus the k x generated by ofr/i is all of Q,u /Hy)^l\j p jP so our claim is trivially true. • 99 <p and Ou, /v\Ou, P P =0 Chapter 3. The Cohomology of M 3.1.2 (P , 2) n 0 j 0 T h e Hypercohomology Calculation We shall first change coordinates on Uo and U\, the reason for this is to make the extraction of the invariant part of the hypercohomology more simple. We choose new coordinates as follows: di - n , Si = n,i + d'i = <i n , < i + = 2 2 <2 We list coordinates on U as follows 0 (m, &i, 6 , . . . , b , s , d , s , d ,... , s , d ) 2 n 2 2 and for t7i, the order is analogous. 100 3 3 n n Chapter 3. The Cohomology of M (P , 2) n 0 > 0 In these coordinates the vector field on UQ and U\ is \ m(-4&i + s ) ( -m'(s' + s' b\ + 2b' ) 2 3 - 4 m - b\ + b 2 - 2 m s - bib + b 2 3 2 -m'((s ) - (d ) ) + b' - (b' f 2 3 2 2 —2ms — bib + ft 3 2 -2m's' + 1 - b' - b[b' 2 2 2 3 2 3 -m'{s' s' -d' d' )-b' b' 4 2 3 2 3 2 2 + b', 3 -2msi - bibi + b x i+ -2ms - bib n sh 2 -m'(s' s' n 2 - | («1 + di) - 2b + s 2 2 2 3 2 2 3 2 n ^ 2 2 n n 2 3 2 3 2 2 2 2 3 2 2 3 2 3 2 3 3 2 3 2 3 2 3 i+l 1 i+l \ (s' s' + d' d' ) + \ (s' s' + d' d' ) b[ - s' b' 3 J n n 3 n 2 n 2 n 2 3 n 2 n 2 n 2 (7i - ^ ( m ' d ( s + 2)), 2 2 abusing notation we shall call this new neighborhood Ui also. We observe that V(m'd (s' + 2 2)) meets the fixed locus in Ui at c = 1, since on the fixed locus, c - 2 „ n I ( 4 < + d' s' ) + \ (s' d' + d' s' ) b[ - d' b' V We now need to find a companion pair for V. Suppose we replace Ui with / 4 4 2 n 2 2 \ K d j + disj) + \ (s'A + d ^) b[ - djftj + d' d b\ - \ {s d + d s ) n 2 \ (s^ + d'^) + I (s' s> + d'A) b[ - s'M + s' i+ 2 3 3 3 i+ n 2 s' cf + i (s' d' + d' s' ) b[ - d' b' + d' - \ {s s + d d ) - 2b 2 3 2 dih - \ (s di + d Si) + d i sh 2 3 3 2 2 n \ « s ) + (d' ) ) + \ (s' s' + d' d> ) b[ - s' b' + s 4 «A ~ \ (s Si + d di) - 2bi + s i 2 3 2 3 3 2 \ (s' d' + d' s' ) + s' d' b[ - d' b' + d' d h - \ (s d + d s ) + tit 3 n 2 2 3 S3W - \ (s s + d d ) - 2b + s 2 2 ± (s' s' + d' d' ) + I ((s ) + {d' f) b[ - s' b' + s' 3 d bi — s d + d 2 - d' d' ) - b' b' n , 2 , n 2(c -1) and c ^ 2. Thus UQ and our new neighborhood Ui still cover the fixed locus. 101 2 n J Chapter 3. The Cohomology of M ,o (P , 2) n 0 Lemma 25 Let Id (W ,Wi)0 d d ddn (s' + 2) d 2 ds' 2 d' n dd' 2 n Then (Wo, W\) is a companion vectorfieldfor V for n > 3. Suppose that we let v be the image of dy under V. Then the proof of Lemma 25 is y dependent on the following conjecture. Conjecture 1 The sequences (v ,v ,v ,... m bl bn (*Wi tty, vy , • • • , vy , 2 ,v ) = J ,v ,v ,v ,... b2 S2 d2 Sn 0 • • • , s' _ ,'"d' _ , (4 + 2)v , - d' v , ) = h v n n 1 n l s n 2 d n are regular sequences for n > 3. This conjecture can be verified up to n = 10 on Maple for the first sequence, and to n = 6 for the second. In general proving that the first sequence is regular amounts to showing that the three term sequence - ^ (s s + d d ) -2b ,m (-4&i + s ), b d - i (s d + d s )j (b\s n 2 n 2 n n 2 x n 2 n 2 n is a regular sequence in the ring C[m,s ,d ,b ] 2 2 1 where s , d and b are defined as follows: n n n ( Sn \ ( - h +U 2 n \ 2 \d -bi + \s 2m 0 2 \b ) V' ( U 2 "2 2 0 6i ) S2 \ d 2 \4m + bl J It is not hard to show that the first two form a regular sequence for all n, so we need to show that the last one is not a zero divisor when we mod out by thefirsttwo. 102 Chapter 3. The Cohomology of M ,o (P , 2) n 0 Showing that the second sequence is regular amounts to showing that in the ring c[m', 4 M^&'g] , 4 4 , 4 the following seven term sequence is regular for all n: -m'{s' + s' b[+2b' ), 3 2 2 -2m's' + 1 - b' - b[b' , 2 3 2 -m>((s' f-(d' )*) + b' -(b' f, 2 2 3 2 \{s' s' + d' d' ) - s' b' + i ( ( 4 2 3 2 3 2 + {d' f)b' + s' 2 2 2 x 3 ^(s' d' + 4 4 ) - d' b + s' d' b[ 4- 4 2 3 2 - r o W „ (4+2) ^ ( 4 4 - 2 2 2 d' d' ) - b' b' , 2 n 2 n + « ) + \{JA + dMK Q(44 = (v, ^, v > , tv, ^, v > , ( s + 2)v > - d v , ) - b B 2 4&;) - 4 d 2 + 4 4 ) + ^ ( 4 4 + 4 4 W s 2 where n 2 d n n-2 ^-iw + ^ i ) dL - § ( 4 ^ + 4&'i) m's -ite+^i) 4 ^ - | ( 4 + 4&'i) 4 b y —m'd' 2 2 2 If we replace Ui with Ui - v ( ( 4 ) 2 - (4) 2 + 3 4 + 2) - n ( 4 f - + 24) again abusing notation we shall call this new neighborhood U\. Then U\ and UQ still cover the fixed locus. We can solve the system above for m', b' , b' , s' , d' . First 2 v m> = 0 => 4 - 103 _ 4 & i _ 24 3 3 3 Chapter 3. The Cohomology of M (P , 2) n 0 > 0 since m! does not vanish on U\. v = 0^b' Vi If we add = -2m's' + 1 - b[b . 3 2 2 and vy and solve for m' we have 2 1 - b[b> - {b' f 2 m 2s 2 + (*' )2-(d )22 •/ _ v ' = 0 ==> d d 2 2 2 d b' — s^d^b'i 2 2 3 + S S 2 2 3 Substituting in our expressions for s' we have 3 d , _ d' (s' b[ - Ab' ) 2 2 2 s' + 2 3 • 2 Finally we can substitute our expressions for d' and s' into v > = 0 and solve for b' , so 3 u_ 3 -b'As' y-(d' f 2 s 2 2 2 2 + 2s' ) 2 2(( ' y-(d' y 2 s 2 + s' s 2 + 2y Thus we are reduced to showing that the pair K ,(s' + K { . 2 n 2 give a regular sequence in C 2> s b\i YT( 4 ) - ( 4 ) + 3s' + 2' {s' f 2 2 2 2 1 1 {d' Y + 2s ' s' + 2' 2 2 2 P r o o f of L e m m a 25 One can clearly see that W{ does not vanish on U\. When n > 3 and Wo is written in Uy coordinates we have Wo = ^-W 2v\ u so W and W\ differ by a scalar on UQ\. That the kernels /Q map injectively under i(V) 0 is a consequence of Conjecture 1 using the criterion in Lemma 26. • . 104 Chapter 3. The Cohomology of M ( P , 2) n 0 ) 0 To show that one can dispense with the terms of the double complex involving for j > 2 , one must show that one has a regular subsequence of length 3n — 2 among the v y on each neighborhood. Thus is true on UQ if one can establish that the first sequence in Conjecture 1 is regular for all n and on U\ is a consequence of the following conjecture. Conjecture 2 The sequence , tv) (v >, U f c i , . . . , v , v > ,v > ,... , v ' ^, m is regular for all Vn s 2 d 2 s n n n>2. Again this reduces to showing that (vy ,v ' ) is a regular sequence in n r L [ 2 ' , ' n i h 2 s ( 4 ) - {d f 2 2 L_ 1" i + 3s + 2 ' ( 4 ) - (cf ) + 2 4 ' s + 2 ' d' _ • 2 2 2 2 2 2 This has been verified using Maple up to n = 6. • L e m m a 26 TTie double complex (*) is quasi-isomorphic to the following double complex when n > 3 (we shall deal with the case n = 2 at the end of this section): o -> 0 0 t t o (u )/i -» (Otfo(MM))<WK) Uo t ol t Jo = ( « m , Ufti, «6a, • • • , « f c „ W , ^ d , • • • , V ) S 2 105 2 Sn 0 Chapter 3. The Cohomology of M , (P , 2) n 0 0 Proof Immediate. • The group Z x Z 2 acts on the double complex above. 2 is induced from the action of Z x Z 2 Ou (Uo)/Io © OuiiU^/h 2 On 0r/ (^oi)/% the action o on r(C/oi) (see Section 2.6.1). For the terms the action is just induced from the Z action on each compo- 0 2 nent of the direct sum. One can check easily that the maps are equivariant. Thus our task at this stage is to calculate the degree zero hypercohomology of the Z x Z 2 invariant part of this double complex (**), we shall use the notation H ° ( * * ) Z 2 > < Z 2 , we shall eventually show that in fact this is equal to the hypercohomology of the complex (*). From this point on we shall use the notation, for % = 2 , . . . , n ( -bx + \s 2 ~h + \ \d di+l S2 2 \ 2m 0 ^ ( -h + ls V" / s 0 d \ 1 2 \d 2 2 2 \ 4m + 6 J bi J 2 and in particular / Sn+l \ d +l b n+l V Sn = n \ ( J V bn 2 -&x + \s \d 2 2 dn \ v \d. «2 2 ) \ 2m \ * S2 0 d bj V 4m + b\ 2 T h e o r e m 6 The C-module homomorphism ij>: <C[g,s M] -+ 2 {0 {U )lh)d d{d )®{0 {U )lh@0 {U )/h) Uo 106 Ql 2 n Uo Q Ul l 2 ) Chapter 3. The Cohomology of M (P , 2) n 0 ) 0 where </>(!) = (0,1,1), 1>(g) = (o^i + iem^ft^ + ^ ^(6i) = 1P(S ) = 2 + ^X^) -^) )), 2 2 (0,6!, 6a), (0,S2,-*3 + 2&2-4&i) induces an isomorphism, ip, ^:C[^,5 ,6 ]//^H r) 0 2 Z 2 X Z 2 1 , where I — ( n + l , /n+1, 7"n+l) s • for ^ S n + i \ / ±s r V ) n-l 0 Y f \ lg /n+1 V n+l 2 -h l S 2 o - _ b l o h s 2 \ 5 + 46? 2 j V &i(* -4&i) ] 2 or equivalently I = (r (&i), s B n Q s 2 - 6^ + ^ / , Q s n 2 - 6^ /„ + ^ s n + 26^ (4&i - and Si and fi and r{ are defined recursively as follows: S2 S2 h g + 4b 2 6 (s -46!) 2 2 fi+l \s 2 - h\ fi +. \gsi + 2b[(4h - s ) 2 Si+1 Ti+l fi bin for i = 3 , . . . , n. 107 Chapter 3. The Cohomology of M (P , 2) n Qfi We shall first show that ip is injective, then surjectivity shall follow from a result of Getzler and Pandharipande on the Betti numbers of Mo o(P ,2) [9]. We shall discuss this n t result in Section 3.2. First two preliminary lemmas. L e m m a 27 Let s is not divisible by s — 4&i. € C[s , h, m, d ] be as above. Then s n 2 2 n 2 Proof We shall show that if we take m - and s = 4&i that s 2 n ^ 0, thus s n by s - 46 . 2 x >2 We show by induction that when s = 4bi and m = - y | 2 = 2(n)b\ln-l s n d = n b n ^ « - 2 = n so in particular s ^ ^ 0. Now 1 s \ 1 ^s J = -s dz - -d b = 2ms + b b = 12mbi + bl =--djbi 3 2 2 \b - + ^{d\ + 16m) + 2b\ = %b\ 2 x (h - s ) = 3d bi 2 2 3 3 2 1 2 + bl and Si+i = Si (-b t + ^s ^j 2 + ^d di 2 ^-1)^2^-2 2(*)&i + -^-4-^^61 2(i + l)b{, 108 + 2bi , / + ( i-Wj2,i-2 7^d b\- + 2b\ z 2 2 is not divisible Chapter 3. The Cohomology of M (P , 2) n 0 ) 0 similarly di+i = di (-bi + ^s ^ + 7^d Si 2 2 -d b\~ + id b\~ l 2 2 2 and \ + l = 2msi + bibi 4m(i)b[- + ^d b[- 1 id 2 1 2 b[ ^j +1 + 2 8 so we are done by induction. • Lemma 28 We have an isomorphism of rings C[s ,h,g]/ (p i, f , r„+i) 2 n+ n+1 C[s ,b d\ + 16m]/((s , d d , b ,m(s 2 u n+1 2 n+1 n+1 2 - Ab )) n C[s , h, d\ + 16m]). x 2 for all n>2. Where /n+l \ \ r +i / n \g \s -b 0 0 2 x S2 -2 g + Ab \ 0 b x 2 J \ b (s - Ab ) ) 2 2 x Proof Consider ( s i , d d x, b i, m(s — 46!)). We shall show first that these generators only n+ 2 n+ n+ 2 109 Chapter 3. The Cohomology of M (P , 2) n 0 > 0 contain even powers of d , by induction. 2 For % = 2 it is clear. If we assume that s*, d di and bi only contain even powers of 2 d . Then since 2 Si+i dd 2 view (s n + i, 2 2 = 2 2 2msi + bibi + 2 b i, m(s — Abi)) as an ideal in C[m, s ,bi, d ,] and intersect this ideal 2 n+ with C[s ,bi,dl + d di) + 2bi d di i and 6j+i only contain even powers of d also. Therefore we can d d i, 2 2 2 i+x Sj+i, - ^ (s Si Sih = d bi - i (s d di + d\sij i+i b we see that = n+ 2 2 + 16m]. 2 We observe that (a„+i, d d i, 2 n+ b +i, m(s - Abi)) = (s +u d d n 2 n 2 + 46 n+x B+1 , b i, m{s - Abi)). n+ 2 We shall show that in C[m,s ,b d ]/(m{s -Ab )) 2 2 /n+i Pn+l u 2 2 = = dd i 2 n+ l + Ab i n+ Sn+1- We use induction on i. For i = 1 we have f = g+ Abl = d + 16m + Ab , and d + Ab = 2 2 2 2 2 d + 4(6f + 4m), and p = s so equality holds. If we assume the result holds for i then 2 2 dd i 2 i+ + Ab i i+ 2 — -d dibi + ]-d (s di + d Si) + A(2mSi + bik) 2 2 2 2 = -bi(d di + Abi) + ^d (s di + d Si) + 8 ( m S j + b bi) = -bi{d di + Abi) + \si{d\ + 16m) + 861 b - 2s h + \s {d di = -bi(d di + Abi) + \si{d\ + 16m) + ^s (d di + 46;) 2 2 2 x 2 2 t 2 2 +(bi - b\)(8bi - 2s ) + 6* (861 - 2s ) 2 2 110 2 2 2 2 + Abi) Chapter 3. The Cohomology of M , ( P " , 2) 0 Now (bi - b{)(8bi - 2s ) = fm(s 2 0 — Ab ) for some / e C[m, s , bi, df,] since bi - b\ is 2 x 2 divisible by m. Therefore in C[m, s ,b ,d ]/m(s 2 d d i + Ab 2 i + x 2 — Ab ) we have 2 x = Q s - 61^ d di + Abi + i+1 2 = + 16m) + b i ( i - 2s ) 8b 2 2 Q s - 6 ^ fi + ^ (dl + 16m) + b[(8h - 2s ) 2 Pi 2 = fi+1Now in C[m, s ,b , d|]/m(s — 4&i) we have 2 x 2 Si = Si+i Qs 2 -b^j +^ (d di + Abi) 2 = Pi+i as required. Thus in C[m, s , b , d^] 2 x (Pn+i, / n + i , - 46i), 6 i ) = (sn+i, d d ra(s2 n+ 2 + Ab , b n+x n+1 n + u m(s - Abi)), 2 and if we intersect these ideals with the subring C[s , b , d\ + 16m] equality still holds. 2 x So for this proof it suffices to show that (Pn+i, /n+i,m(s - Ab ), 6 i ) n C[s ,61,d + 16m] = (p i, / x 2 Since p n + i and / n+ 2 2 n+ B + 1 ,r B + i). i are elements of C[s ,61, d , + 16m] it suffices to show that 2 B + 2 (m(s - 4&i), 6 i) n C[s ,61, d\ + 16m] = ( r 2 B+ 2 B + 1 ). We now replace C[m,s ,61,d ] with the isomorphic ring C[m,<?,s ,&i]. Suppose / £ 2 2 2 (m(s —46 ), 6 )nC[s ,61, rf +16m], then for some polynomials^ and# in C[s ,b ,g, m] 2 x B+1 2 2 2 2 we have f(h,s ,g) 2 = gi(m, b , s ,g)m(s x 2 2 - Ab ) + g (m, b , s ,g)(2ms x Ill 2 x 2 n + b b ). x n x Chapter 3. The Cohomology of M (P , 2) n 0 ) 0 Taking m — 0 we have f(bi,s ,g)=g (0,b ,s ,g)(b ) n +1 2 thus 2 1 2 1 divides / . Taking and taking a common denominator we have s f{bi, s , g) = - (s gi n 2 n (-^> h, s , g^j 7 ^ 0 * 2 - 46i)^ , 2 where this is an equation of polynomials. But by Lemma 27, s — Abi does not divide s 2 so s — Ab\ divides / and / G (b (s — Abi)) as claimed. • n+1 2 n 2 For simplicity we shall write s +i in place o f ' p i from now on, since these polynon n + — 4&i) although their recursive definitions are mials are equal in C[m,s ,bi,dl\/m(s 2 2 slightly different. Lemma 29 ip is injective. Proof We shall first show that the vertical Cech maps in (**) S : (O (U )/I ) 1 U0 Q 0 (d )d(d ) © {OvMVh) 2 (d" )d(d' ) n 2 n (O (U )/I ) U0 01 0 and $2 : O (U )/I Uo 0 © OuMyii 0 -+ have isomorphic kernels. We need to find an / G O^JJI^/h fd' d(d' )%d d(d ). 2 n 2 112 n O (U )/I U0 01 0 such that d d(d ) 2 n Chapter 3. The Cohomology of M 0 ) 0 ( P , 2) n Now in Ui coordinates, _{d' + \ (s' d' -d' s' )) y/u[ n 2 n 2 n U 2 therefore in (Ou^U^/I^did'J, ^dn d(d ) - ./ i\ dd . ,. n n dd„ dd -di, n 7 ds' \s' + 2 n d d(d ) 2 n d' m (W + 2f-{d' Y) 2 = n d(d' ) ddl 2 2 2 4 d(d' ) n +2 8 (4 + 2)(24 + ( 4 ) 2 - ( 4 ) 2 ) , the last equality results when we write all of the variables in terms of s , d! and b[. Thus 2 2 in particular 8 /= (s' + 2)(2s' + 2 (s' y-(d' r) 2 2 2 is invertible and fd' d(d' ) 2 % n d d{d ). 2 n We observe that Z xZ 2 (O (U i)/h) U0 C m,bi,s ,dl, 2 C m / {s bi - - (s s + d d ), n m,bi,dl, 1 2 0 2 n 2 n -2ms n / (s bi - ^ (46is„ + d d ), n m 2 n - b b , m(-46i + s ) x -2ms n where, for i = 3 , . . . , n, S2 = 46i b = 4m + b\ Si+l = - S i h + ^ (4feiSj + d di) di+i = - d i h + i = 2msi 2 2 + (4Mi + bibi. 113 d Si) 2 + 2bi n 2 - Chapter 3. The Cohomology of M , (P , 2) n 0 0 For simplicity we shall let R = C m, bi, d ,, — / (s h - ^ m 2 n (4&!S + n d d ), -2ms 2 n n b b ^j x n Thus we need to show that = -(61) 5 (-s + 2b -s' b ) , 2 , 3 , 2 2 l 6 (dl + 16m) (6W + 4 ) + 4 m ' ( ( 4 ) - ( d ) ) 2 2 2 2 2 2 in R. The first is easy since b' -2ms + 6i6 _ —8mbi + b\ + 4m&i 2 2 b\ — Am b — Am 2 61. For the second, using the transition functions -s' + 2b' - ' b[=Ab 3 2 in R, so it has the same image as s . 2 S 2 l In the last case one writes d + 16m and 2 (b'^ + d' ) + 4m'((s ) — (d ) ) in terms of b[, s and d' using in the transition functions 2 3 2 2 2 2 2 2 and the coordinate conversions given in Conjecture 1, and observes that they are equal. We omit the explicit expressions since they are very cumbersome. Let A — ker#2. Since kertfi = kev8 where the isomorphism is given by the map 2 (gid d(d ),g fd' d(d' )) -> (gi,g ) 2 n 2 2 n 2 we can identify ker^i with A also. Therefore we have homomorphisms A-> (Oei) - > H ° ( * * ) 114 Z2XZ2 Chapter 3. The Cohomology of M ,o (P , 2) n 0 where the kernel of the induced map A->B°(**) Z 2 X Z 2 is the image of A under i(V), thus we have an injective map i/j(y)A^tf(**) . Z 2 X Z 2 We shall now calculate the kernel of the induced map ^:C[g,b s }^A/(i(V)A) u 2 We observe that the image of (gi,g ) £ A under i(V) is 2 (gid v ,fg d' v . ) 2 dn 2 2 d n since (gi,g ) € A corresponds to (gid d(d ),g fd' d(d' ) 2 2 n {g\d v ,gifd' v > ) 2 dn 2 d 2 2 in k e r ^ which maps to n = (gid d i, n 2 gifd' d' ) n+ 2 n+1 under i{V). Thus i/> factors through C[d + 16m, 3 , h]/ ((d dn+i, s +i, b u m(s 2 2 2 2 n which maps injectively to A/(i(V)A). n+ 2 2 2 2 2 n + l ,s n + 1 ,b n + l ,m(s - 4&i)) D C[d + 16m,s ,h]). 2 2 2 But by Lemma 28 we know that this kernel is ( s i , fn+i,r +\)n + C[82,b ,g]/( ,f ,r ) 1 is injective as required. Sn+1 n+1 2 Therefore it suffices to find the kernel of the map C[#,s ,h] A C[d + 1 6 m , s , h ] l ( ( d d 2 - 4&i)) n C[d + 16m, s ,&i]) 2 2 n -> H° (**) * Z2 n+1 • 115 2 So the induced map Z2 Chapter 3. The Cohomology of M ,o (P , 2) n 0 We shall now choose new generators d, c and c for the ring C[s ,bi, g]/I. x 2 2 We shall explain the geometric significance of these new generators in the next chapter. We will show that the generators C\ and c are the first and second Chern classes of the bundle 2 E = 7r*/*Op"(l) where / is the universal map, and 7r is the projection from the universal curve C to A^o,o(P ,2), and the generator d corresponds to the divisor of points (C, /) n such that C is reducible. Lemma 30 Consider the map of rings d? - j + 9 4c? -3^-4* A d + 4ci s 3 d + ci 2 61 Then the induced map C[g, b , s ]/I t C[d, ci, c ]/J = R(n) 2 2 is an isomorphism for each n > 3, where J where g and f 2 2 and r (#n+l) /n+l) = n+l) r are defined by 2 gi d + Aci = d + 16c + 8dci 2 -4c + 2 2 r 2 = d 9 d + ci and / 9n+l ^ d+4ci 3 <*-2ci 6 ( -d +4c? 2 !-2c /n+l Vr n+1 2 -4c d+ci 3 ) 116 d +16c +8dci 9 2 4=f^ 1 / V 2 + d(^tf 2 Chapter 3. The Cohomology of M ,o (P , 2) n 0 Proof Immediate, by taking d = s - 46i, ci = h - s and c = -\g 2 2 2 We would like to calculate the Hilbert series for R(n). + \s 2 2 - b\. • The following lemma shall be useful when we attempt this. Lemma 31 When d = c — 0, 2 /„+i(0, ,0) = a c? Pn+i(0,ci,0) = c*ie?. C l + 1 0 for on i=- 0 /or z = 0,1. For Ci = d = 0 and n even, / n+1 o (0,0,c ) = 0 (0,0,c ) = 2 ( - l ) f cf n+1 2 2 and for n is odd /n i(0,0,c ) = 4(-l) * c * # i(0,0,c ) = 0. + 2 n+ 2 a 1 2 1 2 Proof Clear if one substitutes the appropriate values into the equation t 9n+l ^ z ^ ± i _ /n+1 \ ^n+1 / n-l d-2ci 6 0 2 c 2 -2 6 0 3 • 117 ( d+4ci 3 . . d +16c?+8dci —4C H ^ 2 2 Chapter 3. The Cohomology of M ,o ( P , 2) n 0 Theorem 7 Let d, C\ and c have weights 1,1, and 2 respectively. Then the Hilbert series 2 of the graded C[d, c , c ]-module R{n), x 2 oo equals the polynomial (l-g )(l-g )(l-g" ) n n+1 +2 (l-q) (l-q ) 2 2 Proof First we notice that r n + 1 , f i, n+ and g i have (weighted) homogeneous degree n + 2, n+ n + l and n respectively. For all n, d appears with non-zero coefficient in the polynomial r n+2 n + i . By Lemma 31 n above, for n even c | appears with non-zero coefficient in g x and c n + n+ non-zero coefficient in f +in appears with Thus for n even the dimension of the C-vector space of monomials of total weighted degree k in R(n) is the coefficient of q in the polynomial k (1 + q + q + ... + q ){l + q + q + ... + q )(l + q + q* + . . . + (a )*' ) 2 n+1 2 n 2 2 1 _ (l- "+ )(l-g" )(l-(g )t) 2 +1 2 g (1 - q){l - q)(l - *) ' q n+l For n odd, c appears with non-zero coefficient in / 2 2 n + i and c" appears with non-zero coefficient in g \. So the dimension of the C-vector space of monomials of total weighted n+ degree k in R{n) is the coefficient of q in k (l + q 2 + q + ... + q^)(l + q + ... q^)(l + q + q + ... + (q )^- ) 2 + i _ (i - <r+xi - <?")(! - (g)^) 2 2 (1 - q)(l - q)(l - q ) 2 also. • 118 ' 2 1 Chapter 3. The Cohomology of M ,o (P , 2) n 0 3.1.3 T h e Case n = 2 In the case n = 2 the vector fields on U and U\ are 0 m(s - 46i) 2 - 4 m - b\ + b 2 —2ms — bib 2 2 2 2 2 2 2 2 b\d — s d 0id - s d 2 2 § « 4 ) + K m - 4& 2 2 2 -m'((s' f - (d' f) - (6' ) 2 hs - I (s| + 4) - 2b 2 2 -2m's' + 1 - b[b' 2 y \ -m'(s' b[-r2b' ) 2 2 y J 2 s' d! b\ - d' b' 2 2 2 J 2 We replace Ui with Ui - V(m'd' s ) and as before U and ^ still cover the fixed locus. 2 2 Q Lemma 32 (W W)~( (Wo, ) - 1 5 - m '— 9 m n Wl +f,'+2^ (s 2)—I+5 3 + ( ( g 2 ) 2 + ( d 2 ) 2 ) + 2 5 2 + 2 2+ 9 ^ — J is a companion pair for V. Proof The vector field W written in U\ coordinates is ^W\, so in particular they differ by a 0 scalar on J7oi. We notice that the vector field W\ does not vanish on Ui and in particular its kernel is generated by Thus in order for (Wo, W\) to be a companion pair, we must verify that the sequences (v ,v > ,m'v + (s + l)v >,m'd' v b[ b 2 S2 2 m 2 + Q ((s' ) + (d' f) + 2s' + 2^ v > 2 dl2 2 and (v ,v ,v ,v ) m bl b2 119 S2 2 2 d 2 Chapter 3. The Cohomology of M ,o (P , 2) n 0 are regular. This is easily verifiable using Maple. • We now would like to write down a simple four term double complex quasi-isomorphic to the double complex (*), in the case n = 2. The following lemma is useful in this regard: Lemma 33 Suppose U C A is open, and we have two vector fields on U, V and W, k such that W does not vanish on U, and the equation V-(iW =0 has no solutions except p, = 0 in U x A . If we assume that 1 where w\ does not vanish on U, and then if K is the kernel of W we have a commutative diagram (il„/(i(V)m) + K)(U) ±> A {Ou(U)/(v ,... v ))dz 1 i k (Ou/i(V)K)(U) l 1 OuW/iV! ,v ,--- ,v ) 2 n where the vertical maps are isomorphisms. Proof For each open set V c U the tangent bundle on V is a free Ou(V) module, generated by the for i — 1,... ,k. Since W\ does not vanish on U, we can chose a new basis for the tangent bundle TJJ. Let _d_ _d_ dzi dyi d_ d_ dzi 120 k Chapter 3. The Cohomology of M ,o (P , 2) n 0 In these coordinates W\ = and V = vi— + > The subsheaf of I «i Wi —. corresponding to the kernel of Wi, JC, is generated by dz , dzz,. 2 as a sheaf of 0\j modules. Now O /i(V)K = Oul < i(V)dz ,i(V)dz v 2 Ouf > k [v - — w ,... ,v - —w 2 2 n k = Ov[p]/{vi - fiWi, v - pw ,... ,v - pw ). 2 2 k k But since V — pW = 0 does not vanish in U x A except when p = 0, if we take 1 /=(«!- ftwi, v - pw ,... ,v - pw ) 2 2 n n and V(I) = X, then the natural projection p: UxA -> U x 1 a; x ;u {0} -i> — > • i x O — > • 1/ £ induces an isomorphism p: X-»p(X). Therefore the induced map Ou/{vi,v ,... 2 ,v ) -> Oir[A*]/(vi - A*«;i,U2-/itw , ••• n 2 is an isomorphism also. Therefore cwi(v)/c = Ouj{v u v ,..., ) 2 121 Vn s <Vi(v)0V). ,v -pw ) n n Chapter 3. The Cohomology of M ,o(P , 2) n 0 So the map Ou/iiV^iy^Ov induces a map toul{i{V)($ll)+K) % Ou/i{V)K ] ; ll A nu/(i(Y)(%)+K) O /i(V)(Qu) v where the vertical map on the right is an isomorphism. Thus i(V) is the zero map also. We also have, using Lemma 26, that (Qu/iiiVWl) + IC) (U) * (o (U)/ (v - ^-w , ...,v - v 2 *(Ou(U)/(v ... 2 n ^™*) ) dz x ,v ))dz u k v • L e m m a 34 Let U\, V, and W\ be as above. Then V - pWi has no solutions inU x A 1 except p = 0, where U = m- V(4(s> ) - 2(4) (d' ) + ( ' ) + 4(s ) + 2(d' ) + (4) - 4s' (d ) )2 2 2 2 4 2 3 S 2 2 2 4 2 2 2 2 Proof We shall show that the system of equations -m'{s' b\+2b' ) = -fim' 2 - 2 m V + 1 - b\b' = 2 2 \ ((4) 2 2 2 0 (3.2) = 0 (3.3) 2 -m'((s' ) -(d' ) )-(b' ) 2 (3.1) 2 2 2 + (4)) K - s' b' = p(s' + 2) 2 2 2 (3.4) 2 , + 122 + ( , 5) Chapter 3. The Cohomology of M , (P , 2) n 0 0 has no solution except p — 0 in U, which incidentally contains the points ( - | , 0,0, - 1 , - 1 ) , (—|,0,0, -1,1), which correspond to c = 0 on the fixed locus. Adding jr times (3.5) and times (3.1) and solving for b[ we have, (4 + 2(d ) + ( ) + 4s ) 2 M 2 2 g 2 2 Subtracting jr times (3.5) from — ^ times (3.1) and solving forfr we have, 2 /i(4-K ) + (5 ) +4 ) 2 2 2 6 2 - 2 g2 # ' ( } Substituting (3.6) and (3.7) into (3.4) we have Thus ju = 0 on (7 as required. • Suppose we replace U\ with U from Lemma 34, abusing notation we shall still call this neighborhood Ui, then we have the following: L e m m a 35 Let {v ,v ,v ,v ) m bl b2 = J S2 ( v , ^ , ^ , ^ , ^ 2 ) 0 = Jl. The double complex (*) is quasi-isomorphic to the following double complex (**) in the case n = 2; 0 ->• 0 0 t t -+ (0 (Uoi)/Io)d d(d ) Uo 2 2 0 (Uoi)/I Uo t 0 t -4 (o (c/o)/Jo)(rf2)rf(rf2)e(o (c/ )/J )(i(m ) , t/o 0 f/l 1 1 123 -> o r / 0 ([/ )/J e 0 0 0^(^)7A -> Chapter 3. The Cohomology of M , ( P , 2) n 0 0 Proof Follows from Lemma 31. • We shall restrict to the Z x Z invariant part of the double complex above. In this 2 case v bl 2 = 0 =>• 6 = b\ + Am, and v 2 = 0 => d\ = 2s 6i - s , - Abi ~ lfim, and 2 S2 2 Ouo(U )/Io = C[m,s ,h]/(v ,v ). 0 2 m b2 For Ui, v i = 0 and rf ^d' = 0 =>• b[ = b' = 0 since (d' ) , s and m ' are invertible. Also 2 m 2 vy = 0 =>- m! = ^ r , and x 2 2 = 0 2 2 (d' ) = (s' ) . Thus 2 2 2 2 0 {U )/I Ux x =C 2 >1 s Theorem 8 The map if> : C[s , h] -> (0 (U i)/Io)d d(d ) © (0 {Uo)/I © 2 Uo 0 2 2 Uo 0 Ov^/h) where </>(l) = (0,1,1) ^(61) = (0,6i,0), ^ 2 ) = (0,s ,0), 2 induces an isomorphism, ip, from C[s ,6i]/7 to u ° ( * * ) 2 2 where z 2 x Z ) I = (6? (s - Ah), (s - 26 )(s - s h + 46 )) 2 2 2 2 1 2 We shall again show injectivity and surjectivity shall follow from Getzler and Pandharipande's result in [9] on the Betti numbers of A f Theorem 9 ip is injective. 124 ( P , 2 ) , we discuss this in Section 3.2. n 0 ) 0 Chapter 3. The Cohomology of M , ( P , 2) 0 n 0 Proof As in the proof of the general case, we first show that the vertical Cech maps in (**), 1 <*i : (C[m, s , bx]/{v , v )) (d )d(d ) © C2 ' j 2 m b2 2 dm' S 2 ^{0 (U i)/Io)d d(d ) Uo 0 2 2 and <1 S :0 (U )/Io®C 2 Uo ^O (U )/I s 0 U0 01 0 have isomorphic kernels. This will follow once we show that / Si d dm' A- —d(d ), 2 2 Now Si(dm') = ^d(d ) dd 2d m 2 2 2 4(6? - 4m) m'u 2 Ui d(d ) 2 d d(d ) 2 2 where the last equality comes from writing the previous expression in Ui coordinates. In the ring Ou^U^/h, m' = u = s' + 1 and u = 2 2 so x 2 2 m'u _ 1 Ml 4 2 Thus S^drri) = %d(d ). 2 Let A denote the Z x Z invariant part of the kernel of S. Then there is an injective 2 2 map 0®A/{i(V)A) -*.H°(**) Z 2 induced from the inclusion 0 © A - » (O (U )/I ) U0 01 0 d d(d ) © (0 (U )/Io © 2 2 125 Uo 0 0 {U )/h). Vl x Chapter 3. The Cohomology of M (P , 2) n 0 ) 0 Therefore we need only show that ip(C[bi,s ]) C A and the kernel of the induced map 2 C[b s ]^A/(i(V)A) u 2 is I. To show that ip(C[bi,s ]) C A we must show that <5i(&i,0) = Si(s ,0) = 0, i.e. that 2 s = 2 O l = 0in 2 O (U )/I . U0 01 0 First we note that m and 12m + b\ do not vanish on the fixed locus in Uo except at the point F ^ , thus (replacing UQ with a smaller neighborhood if necessary) on U i we 0 can invert these functions. Since h = \ m(s - 46i), - 4 m - b\ -I- o , -2ms - bib , b s - - (s . + d\ 2 2 2 2 2 x 2 in 0 {Uoi)/Io we have Uo s = 46i 2 4m + b\ and - 2 m s - 6i& = 0 =• -6i(12m + b\) = 0 =• &i = 0. 2 2 Thus S2 = 6i = 0 as required. We shall now calculate the kernel of the induced map iP:C[b s ]^A/{i{V)A). u 2 First we observe that the image of (gid d(d ), g dm') G A under i(V) is 2 2 2 {gid v ,0) 2 d2 126 Chapter 3. The Cohomology of M , (P , 2) n 0 0 Thus i\> factors through C[m, s ,61]/ ((d v , v , v ) n C[m, s , h]) 2 2 which maps injectively to A/(i(V)A). 4 C[s ,6i] 2 b2 m 2 Thus it suffices to find the kernel of the map C[m,S2,6i]/(d2««fe,U6a,t;m) n In terms of m, s and 61, d ?j 2 d2 2 rf2 (C[m,s ,b ]). 2 1 = (2s 6i - s\ - 46? - 16m) (61 - s ) , thus we need to find 2 2 the kernel of the map C[s ,&i] i C[m,s ,6 ]/((2s 6i -s\- 2 2 1 46? - 16m) (6 - s ), -6 (6? + 12m),m(s - 460). 2 X 2 X 2 Now ((2s &! - 4 - 46? - 16m)(6i - s ), -6 (6? + 12m), m(s - Abi)) 2 2 X 2 = ((2a 6i - s| - 46?)(6 - s ) + 48m6i, -61(6? + 12m), m(s - 4&1)) 2 X = ((2s 6 - s \ 2 x 2 2 AbDih - s ) - 46?, -b {b\ + 12m), m(s - 46 )). 2 x 2 x Thus we need to find the kernel of C[s ,61] -> C[m, s , 6 i ] / ( - 6 i ( 6 + 12m), m(s - 46i)). 2 2 2 2 Suppose / is in the kernel then f(s , b ) = -gi(m, 2 x 8 ,61)61(6? + 12m) + g {m, s , b )m(s 2 2 2 l 2 - 46i) for polynomials g and # . Taking m = 0 we see that 6? divides / , and taking m = —^2 x 2 we see that (s — 46i) divides / , thus / € (6?(s — 4&i)), so the kernel contains the ideal 2 2 (6?(s — 46i)). Now 6?(s — 46i) lies in the kernel since 2 2 bl(s 2 - 4&i) = - ( s 2 - 46i)6i(6? + 12m) + 12m(s - 46i). 2 Thus the kernel equals the ideal 6? (s — 46i). Therefore the kernel of the map 2 is the ideal {b\(s - 4&i), (2s 6i - s , - 46?)(6i - s ) + 46?). Now the polynomial (2s 6i - s 2 2 2 2 127 2 2 - Chapter 3. The Cohomology of M ,o (P , 2) n 0 46?)(61 - s ) + 46? factors and equals (s - s b + 46?) (s - 26i) as required. • 2 2 2 x 2 In this situation we also choose new generators d and C\ for the ring C[6i,s ]/7 where 2 d = s — 46i, ci = 61 — s . Their geometric significance is explained in the next chapter. 2 2 In terms of these generators, I = ((d + c fd, (Ad + l l d c i + 16c )(d - 2ci)). 2 2 x If we let R(2) = C[d, ci]/((d + ci) d, (Ad + l l d c i + 16c )(d - 2ci)), 3 2 2 where d and C i each have weight 1, then R(2) has Hilbert polynomial P(R(2), ) = (l + q q 2 q + )(l 3 + q (1 q) = 2 + q + " f^~ fl ^ ( ( g4) ( 3.2 The Cohomology of M , ( P , 2) n 0 0 First we mention an important result. L e m m a 36 H (M (P , q n 0fi 2), W) = 0 when p^q. Proof One can deduce this fact using the methods of stratification of A^o,o(P") 2) as in [9]. • As explained in the introduction, the result above is used to show that H* (M (? ,2)X) n sing Qfi = Gr (1$(M (V\2),1C)) F Qfi as graded rings where the filtration F comes from the spectral sequence. But the result of Akyildiz and Carrell [1] mentioned in the introduction tells us that we do not have to 128 Chapter 3. The Cohomology of M ,o (P , 2) n 0 calculate the graded ring associated to this filtration since H° [Mo,o{P , 2), K.*) is already n graded (this grading is given by the action of G ), so m if* (M ,o(P", 2), C) S B°(M {P , 2), KT) n n9 0 0fi as graded rings with respect to this grading. In particular the Hilbert polynomial of the graded ring H ° ( A T , o ( P , 2), £ * ) equals the Hilbert polynomial of H* (M ,o( > n )> )- pn 0 sing 2 0 C Getzler and Pandharipande [9] show that the Poincare polynomial of Mo o(P , 2), n t e(M),o(P ,2)) = n ^(-t)Mimfr(Mo,o(P ,2),q n i (1 - q )(l - q )(l n - q ) n+l n+2 (1 - qf{l - q*) ™ { } where t = q. 2 We shall show that the induced G action on the subring R(n) of H ° ( A f o , o ( P , 2 ) , f 2 * ) n m has weight respectively 1,1, and 2 on the generators d, C\ and c . We calculated the 2 Hilbert polynomial of R(n) with respect to these weights and found that it was equal to the polynomial in Equation 3.9 above. Thus R(n) = H ° ( ^ t o ( P , 2 ) , / C * ) = H: (M (P ,2),C) n n 0! ing 0fi as graded rings. Therefore we have the following theorem. T h e o r e m 10 Let d, c and c have weights 1, 1, and 2 respectively, then x 2 R(n) = C[d, , c ]/J = # * ( M o ( P , 2)C, C) n C l 2 ns 0) as graded rings, where J — (^n+lj /n+lj n+l) r 129 Chapter 3. The Cohomology of M (P , 2) n 0 > 0 and / ^n-1 d-2ci 6 9n+l ^ d+4ci , -4C /n+l V V r„+i J 0 d+ci 3 0 . d +16c?+8dci 2 2 H ^ — V Proof We need only show that the induced G action on R(n) gives the generators d, C\ and m c weights 1,1, and 2 respectively. The result of Carrell and Akyildiz in [1] mentioned 2 in the introduction gives us that H° (.Mo o(P 2), Q*) is graded via the action of G and n ) J m the filtration F is the canonical filtration associated to this grading. In particular the 9 induced G action on R(n) (which can be viewed as a subring of H°(.Mo,o(P \ 2), Q,*)) 7 m gives a filtration of R(n) as follows: F (R(n)) = {/ € R(n)\X*f = X~ f. Q kq Since R(n) = C[d,c ,c }/J 1 it suffices to calculate the weights of the G m 2 action on the generators. Lemma 14 we have X*m X~ m X*h X'% X*s X~ s 2k k 2 2 x- d?. k X*d 2 Thus since d Cl c = = 82- 46i b i - 8 ~(d 2 2 2 + l6m) + 130 ±s -b , 2 2 Now from Chapter 3. The Cohomology of M ,o(P 0 G m acts with the required weights. • n Chapter 4 Geometric Interpretation Let U — {Ui} be the etale cover for Mofi(P , 2) we constructed in Chapter 2. Each Ui n equipped with universal curve Ci and universal map fi. Consider the following diagram: f*0(l) -> 0(1) d 4 p n Ui We can push forward the bundle /j*0(l) to get a rank three vector bundle onUi,ir*f*0(l). Thus gives a vector bundle E on Afo,o(P", 2). In this section we shall calculate the Chern classes of this bundle, in terms of the generators of the cohomology ring of M , o ( P , 2). n 0 We shall show that the cohomology ring can be generated by the first and second Chern classes of the bundle E and one other degree one generator, which is a divisor consisting of the points ( C , / ) 6 M o ( P , 2 ) such that C is reducible. We shall give the relations n 0) in terms of these generators. We can also compute the Chern roots of E, 71, 72 and 73 of E. In Section 4.6 of this chapter we give the cohomology ring as the invariant subring of Cpyi, 72,73]/(i?i, R2, R3) under an action of Z , where Ri, R and R are invariant polynomials under this action. 2 2 3 132 Chapter 4. Geometric Interpretation For arbitrary n we shall show how one can describe some other divisors with geometric significance in terms of the generators s and b (Section 4.3), in particular we shall con2 x sider the divisor corresponding to pairs (C, / ) where f(C) intersects a fixed codimension two linear subspace of P . This divisor will correspond to — s in our cohomology ring. n 2 In the case n = 2, M o ( P , 2 ) is the space of complete conies, which is isomorphic 2 0) to the blow-up of P along the Veronese Surface, P . We shall construct an isomorphism 5 5 from the scheme XJ§jTL to an open subscheme of P . We shall also calculate the coho5 2 mology of P in terms of the hyperplane class in P , H, and the exceptional divisor T , 5 5 and give the relationship between these generators and the generators s and b\. 2 4.1 Chern Classes In this section we shall define the Chern Classes of a vector bundle E. But first some preliminary definitions. Let X be a smooth variety and E a vector bundle on X. Definition 14 A connection is a homomorphism of sheaves of C vector spaces V : E -¥ satisfying Leibnitz' rule, i.e. fix ® E if U is an open subscheme of X, e e E(U), then V(U)(fe) = df®e + fVe. L e m m a 37 Let V and V be connections, then V- V : E -> Q x 133 ®E and f G 0(U) and Chapter 4. Geometric Interpretation is Ox-linear. Thus is a homomorphism of vector bundles. Proof Let / G 0(U) and e G E(U), then (V - V ' ) / e = (df ® e + / V e ) - (d/ ® e + / V e ) = / ( V - V')e. • Definition 15 Suppose thatU = {Ui} is a collection of locally trivial open neighborhoods for the rank h vector bundle E, i. e. E\u = (Q ) h t Vt then we can define a family of local connections {Vj} as follows: Vi:O h Vi A Q ®O h Ut (/i, ••-,//>) ->• {dfi,... = Q h Vi Vi ,df ). h Definition 16 Let (Ui, Vj) 6e /oca/ connections. Then c(E) = (Ui ,Vi-V )eH\X,Uom(E,n ®E)) j j = x H\X,Uom{E,E)®Sl ) x is called the Atiyah class of the vector bundle E, and is independent of the local connections chosen. Let M(k, C) be the variety of k x k matrices over C. We introduce now the notion of invariant polynomial due to Carrell and Lieberman [6]: Definition 17 Suppose p is a polynomial map p:M(k,C) 134 ->C Chapter 4. Geometric Interpretation such that p(N~ AN) = p(A). Then p is called an invariant polynomial of degree d if 1 there exists a C-linear map 4>:M(k,C) ® . . . ® M ( f c , q - > C v v ' d times such that the following diagram commutes: M(k, C) C A \ ^<f> M(k, C) <g> . . . <g> M(k, C) d times tu/iere A is the diagonal map. For example, suppose X +p X k k 1 1 + ...p . X+p k 1 k = 0 is the characteristic polynomial of the k x k matrix A = {a ,}. Then each of the coefi3 ficients Pi is a degree i polynomial in the and is an invariant polynomial of degree i (see [6]). Suppose E has rank k. Let A G End(E), and let {Ui} be a collection of affine open neighborhoods such that E is trivial on each C/j. Then A is given on Ui by a matrix Ai G M(k, 0{Ui)) and if b : E\ -4 Ut (0 ) k Ut then thus Ai and A,- have the same characteristic polynomial. We define the characteristic polynomial of the endomorphism A to be the characteristic polynomial of any A^ 135 Chapter 4. Geometric Interpretation Let p (A) d be the degree d coefficient of the characteristic polynomial of A 6 End(E). Then as above we have a commutative diagram of sheaves: End(E) O x End(E) ® ... ® End{E\ d times where p is Ox-linear. This gives us a commutative diagram d End(E) <g> n (fi*)® A \ / * p (8> irf ®d x d d n (End(E) ® fix) ® • • • ® (End(E) ® fix) " v ' d times Definition 18 I e £ 0(75) G H (X,End(E) l ® fix) &e i/ie Atiyah class, then the image of p {c{E) U • • • U c(E)) in H (X, fi^ ) is called the d d rf d t/l C/iern Class, where p d is the degree d coefficient of the characteristic polynomial. 4.2 Chern Classes and Hypercohomology In this section we shall show how to find elements in H ° ( A t , o ( P , 2), JC*) corresponding n 0 to the Chern classes of the rank three vector bundle E on M ,o(P , 2), given on U by n 0 t TT./;0(1). Let V be a vector field on M (P ,2). n M Qfi Let JC* be the complex associated to the 136 Chapter 4. Geometric Interpretation vector field V on A4 ,o(P , 2). Consider the double complex End(E) <g> K*: n M 0 t ••• -> ® End(E) ••• ->• t ® Q(U ) i<j<k @ End(E) ijk ® End(E)^Cl(U ) ••• i<j ijk ® jEnd(E)®0{U ) ij ®iEnd(E) ® 0(U ) i<j<k i< <g> Q(Ui) -> -> 0 ij ® End(E) { -+ 0 ® 0(Ui) ->• 0 with differential id <g> Z(VM)- Consider the Atiyah class c(E') G i / ^ M ^ P " , 2), End(E) <g> fi), then c(£?) can be represented by an element of ® jEnd(E) i< ® Sl{U%j) which we shall call c(E) also, with the property that 5(c(E)) = 0 where S is the Cech differential. Now 5{i(V )c(E))=i(V )(6c{E)) M thus there exists an element a G M ®iEnd(E)®Ou (Ui) =0 such that { = 5(a). Then I(VM)C(E) the pair /3 = (c(E), a) is an element of H°(iMo,o(P , 2),End(E) ® /C*). Thus the Atiyah B class can be viewed as an element of H° (A^o,o(P , 2), End(E) <g> £ * ) once we find for it a n suitable partner a. Suppose we can find such a partner a. Then we can compute the image of the pair P = (c(E), a) under the map p d p~ :T$(X,End(E)®1C) -4 d (3 ^ H°(X,/C*) p (pu---Ul3) d d times and the image will be the d th Chern class of E as an element of H°(A fo,o(P / 137 n ) 2), JC*). Chapter 4. Geometric Interpretation Definition 19 Let V:Ox^ O x be a derivation. Let E be a vector bundle over X. Then a lift of V to E is a homomor- phism of sheaves of C-vector spaces :E^E V such that V(fe) = V(f)e + fV(e), for all f e 0(U) and e e E(U). Consider our original vector field on P n induced by the A 1 action M on P n (given in Chapter 2). We have an induced a vector field on .Mo,o(P ,2) and thus a differential n V M on -M ,o(P™, 2). Let E be as before, i.e. E is given by 7r*/*0(l) on Ui. We shall 0 show that the vector bundle E is trivial on Ui. We shall show also that we can find a lift V M '• E —> E. Once we have established these facts the following Lemma gives us a partner a for the Atiyah class c(E): L e m m a 38 Let a e ®iEnd(E) <g> 0 (Ui) Vi a = {Ui,{id®V )(Vi) M - VM) then c(E) = (U V -V ) ijt i -> j (U&iidQiiVMNVi-Vj) {u {id®v )iyi)-v ) h M M so (c{E), a) € H° (M (P , 2), End{E) <g> tC). n 0>0 Proof We just have to show that (id® V )(Vi) M - VM € End(E)(Ui). 138 Let e € E(Ui) and Chapter 4. Geometric Interpretation feOutiUi). Then {(id®V )(Vi)-VM)fe M = id®V (Vi)(fe)-VMfe M = id® V (e ®df + / V M i e ) - (e <g> V (f) M = (id®V )(fVie)-fVM{e) = f((id®V )(V )-V^(e)) + fVu(e)) M M = ie f(id®V (Vi)-VM)e M as required. • Lemma 39 The vector bundle E is trivial on Ui and has basis ms , st and t 2 2 on UQ. Proof We shall show that the vector bundle is trivial on Uo and since the universal maps /o and fi differ only by a projective change of coordinates for P n it follows that the bundle is trivial on each Ui. First we shall show that if we restrict E to the subvariety m = 0 in Uo, E has a basis | , 1, | , which extends to give a basis for E on UQ- Consider f*0(l) on C. Since /(C) C U Xo UU Xl we need only consider 0(l)\ uu UxQ xl On U , 0(1) has local trivializations Xi &:0(l)k 0 Vi i Thus for the relevant neighborhoods, U 1 and U , we have gluing data, ^ I ^ Q = f^1 Xo Xl 139 Chapter 4. Geometric Interpretation Using the notation from Chapter 1, let a = (m, 6 ,... , r ^) € Uo with m = 0. Then the X fiber C a n is isomorphic to {< s, £ > x < p, q >£ P x P | pt = 0}. 1 1 and the fiber of TT*/*0(1) at a is # ° ( C « , / * 0 ( 1 ) ) . Consider the following diagram: s,t> < \ C a A P n <p,q> The induced map from each copy of P to P is an isomorphism. 1 n The map from the top copy of P to P is 1 n < 8,t > ^ < 8,t + biS,... > and from the bottom copy is <p,q>^<q,p + b q,... l > . Thus a section of f*0(l) on C is given by a section of i)f*0{\) on P for j = 1,2 such 1 a that these sections agree when p = t — 0. The vector bundle i*/*0(l) has trivial neighborhoods U = P - < 0,1 >= 1 0 and C/x = P - < 1, -bi >= i7 f~ U , 1 l l Xl and gluing data G+M) ' 140 i7 f~ U 1 1 Xo Chapter 4. Geometric Interpretation so a global section of i\f*0(l) is given on UQ by a s + ait 0 ao,ai6L s and on Ui by OJOS + ait t + bis Therefore H (P ,ilf*O(l)) 0 1 has a basis which is given on UQ by s The vector bundle i* f*0(l) has the same trivial neighborhoods, and gluing data 2 Q ,P + biqJ ' so a global section is given on UQ by Q and on Ui by Ax? + PIP p + hs Therefore ^ ( P , i\f*0(l)) has a basis which is given on UQ by 1 Q Now these sections must agree when p = t = 0, thus a = (3 . Since ij(Uo) C U , 0 0 j = 1,2, the vector bundle /*0(1)) has basis on C fl i7 , which extends to a basis for H°(C , f*0(l)) as claimed. a g a 141 q Chapter 4. Geometric Interpretation We must now show this extends to a basis for E on UQ. Let a = (ra,b\,... ,r ) G UQ, nt2 with m / 0 . Then C is isomorphic to P and is contained in 1 a U UU . p t Consider < s , £ > A C -4< si,ms + hst + t , ...> . 2 2 a Let t / = r /o 1 / " 1 ^ = P - V(st) and C7 = r 1 1 7l / " ^ 1 = P - V ( m s + 6ist + r ). Thus 1 2 2 when we pull 0(1) back to P we have two trivial neighborhoods, Uf and Uf and gluing 1 0 x data ( 5* \ms ) + b st + t J 2 2 1 and a global section of i*f*0(l) is given on Uf by / a s + ctist + a t \ 0 2 2 Q 2 V 7t J and on Uj by x / OJOS + 2 a.\st + a t \ 2 2 V ms + 6ist + i 2 )' 2 Thus # ° ( C , /*0(1)) has a basis given by a s_t_ t s on Uf . However since m ^ O w e can choose as a basis 0 l,m-,-. t s We have five trivial neighborhoods on C , namely the intersections of f~ {U ) and each l Xi of U , Up and U , for i = 0,1. We have five instead of six because f(U ) C U . We call t q q them U , U , U , U and U , where t/ = / ( ^ x ) _1 to tl Po P1 q to n 0 etc. The neighborhood C/ = { ( | , m , . . . , r , ) G I7 | s # o} t0 n 142 2 t xo Chapter 4. Geometric Interpretation contains the image of U/ under i. Thus we need to find the gluing data for U and U . Q 0 Since both of these neighborhoods are contained in f~ (U ), to which is a locally trivial l Xo neighborhood for /*0(1), the gluing data is trivial so we just need to write the basis q s in terms of the basis l,ra-, - . *' t We use the coordinate conversion given by the equation t p s mq = -p =>• - — m-. s q t Thus these bases are the same, so in particular the basis 1, | , | extends to give a global basis for E on UQ as required. • We shall now deal with finding a lift of the derivation V on .Mo,o(P ,2). n M First we shall find a lift of the derivation V on P , V , to 0(1), and this will induce a derivation n VM on 7r*/*0(l) = E, which is a lift of VM- Lemma 40 Let V be the derivation V : Op» -4 Op» associated to the A action M on F . Consider the C-linear map from 0(1) to 0(1) 1 1 given by d V : f(x ,... ,x )->— (f{t.(x ,... , x ))) 0 n where t.(x ,... , x ) = M(t)(x ,... 0 n 0 Q n t = 0 ,x ). Then V is a lift of V. n Proof It is easy to see that V is a derivation, so we must just show that it is a lift for V. Let 143 Chapter 4. Geometric Interpretation f G Op»(U), and e G 0(1)(U). Then V{fe) = ^f(t.(x ,... ,x ))e(t.(x ,... ,x )) t=o 0 n 0 d n d t=oe + —e(t.(x ,... ,x ))-of 0 = ^ / ( t . ( x . . . ,x )) = V{f)e + fVe 0 > n n since by definition, if / G Op"(£7), then V(f) = d -(f(t.(x ,...,x ))) 0 We note that in particular Vfa) = x i+i n t=0- for i = 0 , . . . , n - 1 and V(x ) = 0. Since n 0(l)(U) is generated by the global sections 0(C/)-module, the lift is completely determined by its action on these global sections. • This derivation on 0(1) induces a derivation VM on 7 r » / * 0 ( l ) = E, which is a lift of VM since VM is induced by the derivation V on P . We need to find this induced derivan tion V M - We shall calculate V only on C/ since this is the important neighborhood M 0 when we calculate the hypercohomology. Since 0(1) is generated the global sections X o • ' . j Xfi cLS £111 sheaf of O p « - m o d u l e s , E 5 , f*(x ). Thus the lift VM is determined by its action is generated locally by f*{x ),... 0 n on the f*X{. Lemma 41 The endomorphism id® V M ( V O ) — VM with respect to the basis ms , st, t 2 of E on UQ is given by the matrix C where ( C = \ r,i 2 1 0 2m b\ 2m 0 1 - h 144 r 2)2 \ -bx J 2 Chapter 4. Geometric Interpretation Proof Since and the universal map is given by blowing up the rational map (m, b ••• , r fl) X < S,t n < st, ms + hst +1 , mr ,is + b st + r , t ,... , mr s 2 2 2 2 2 2 2 + b st + r , t > 2 2 2 ntl n n 2 we have V {st) = ms + hst +1 2 (4.1) 2 M V (ms + hst + t ) = mr s + hst + r t 2 2 2>1 V (mr is + hst + r t ) 2 M (4.2) 2 (4.3) 2>2 = mr s + hst + r t 2 2t 2 2 M 2 2>2 3]1 3>2 Thus 4.2 gives us V (ms ) + VM(t ) 2 2 M - V^(M*) = mr ,is + b st + r t 2 2 2 2 2>2 = rnr s + b st + r t - V {h)st - h(ms + hst +1 ) = ( r , i - h)ms + (6 - V (h) - b\)st + ( r 2 2 2A 2 2 2<2 2 2 2 M 2 - h)t 2 2>2 M and 4.3 gives V (rnr s 2 M 2tl +r t ) = mr s 2 3tl 2 M 2 2 3 2 M 2 3i2 = mr s 2 2>2 t - V (b st) 2 3)2 2 3)1 =>- r ,iV (ms ) + r V {t ) M 3 mr s + b st + r t - V (h)st - b (ms + hst + t ) = 2 + b st + r 2 2j2 3tl M + b st + r t - V {b )st 2 3 3>2 M 2 -b (ms + hst +1 ) - V (r i)ms 2 2 M (fs,i ~ h - V (r ,i))ms +(r-3,2 - h 145 2 3 V (r , ))t 2 M 2 2 2 2t M 2>2 + (b - V (b ) - hb )st 2 M - V (r )t 2 2 = 2 2 M 2 2 Chapter 4. Geometric Interpretation Now V (h) = M -4m-6 V {b2) = +6 2 2 -2m(r ,i + r , ) - M 2 2 2 (r i) = 6ir i - r VM(r , ) = 6ir , - r\ -b 2 > 2 2 2 ] 2 2 j l 2 2 hb + 63 2 - 6 + r ,i 2 2 3 +r 3>2 , so VW(ms ) + VW(t ) 2 2 r , i i ^ ( m s ) + r VM(t ) 2 2 2 2>2 = ( r i - h)ms = (r ,i(r 2 2) 2 + {4m)st + ( r - 6i))ms + (2m(r 2 2)1 2]1 2)2 - 6i)t + r ))st 2|2 +(r (r ,2 - h))t . (4 2 2)2 Subtracting (4.5) from r 2 ) 1 2 times (4.4) we have ( r , i - r , ) V M ( i ) = 2ra(r 2 ,i - r 2 > 2 )s* + ( r , i - r 2 , 2 ) ( r 2 i 2 - o ^ r 2 2 (4 2 2 2 2 2 Thus V (t ) = {2m)st- (r -b )t . 2 2 M Similarly, subtracting (4.5) from r V (ms ) 2 M 2 ) 2 r 2t2 1 times (4.4) we have = ( r i - bi)ms + 2m(st). 2 2) Thus the matrix is as required. • To find the Chern classes we must calculate the characteristic polynomial of the domorphism given by the matrix C. 146 Chapter 4. Geometric Interpretation Theorem 11 The Chern classes of the vector bundle E = 7r»/*0(l) are ci = -s + h 2 1 2 1 ,2 C2 = - 4 ^ + 4 * 2 C3 = 61 (-c - 61 ci 2 6?) Proof We calculate the determinant of a;/ — C. Now ^ ar xI-C= r , i + 61 — -1 2 -2m \ x-bi 0 N 0 -2m - 1 z-r 2>2 + bi / So the characteristic polynomial of C is z+(-Ki+r,)+&i):z+(r^ 3 2 22 2 ! l and ci = -(r,i+ r )+ 61 2 c 2 = r c 3 = -r 2 > 1 r 2 ! l 2)2 2 > 2 r - 4m - 6? 2 ) 2 6i +(r 2 1 +r ) 6 - ^ + 2m(r i + r 2 2 ) 2 147 2i 2 ) 2 )-4m6i. Chapter 4. Geometric Interpretation In terms of the coordinates of the cohomology ring we have Ci c 2 - - s + 61 2 = \{s\-dl)-Am-b\ 1 c 3 1 2 ,2 - dl)bi + s b\ -b\ + 2ms - 4 m 6 i = ~(sl = 61 = 61 ( - C 2 - 26 + s 6i) = h ( - c - 61(61 - = h ( - C 2 - 61C1 - 6 ) 2 - 2 + s 0i - b 2 2 2 2 2 - b\) 8) 2 2 as required. • 4.3 Some Divisors on A4 ,o(P , 2) n 0 Suppose a divisor D on A to,o(P",2) is given by the local equation fi = 0 on Ui, where / {Ui} is our etale open cover for Mo,o(P , 2). Let L(D) be the associated rank one vector n bundle. Let V* be local connections as before. Then in this case on Uij, v - v * ' " f Thus the Atiyah class c(L(D)) e H^MoA?", Now 148 " /* fi' )> End(L(D)) ® £ * ) is 2 Chapter 4. Geometric Interpretation Therefore, as long as G 0 (Ui), a = (Ui, - ™ ) G ®iEnd(L(D))®0(Ui) Vi is a partner for the Atiyah class c(L(D)) in the sense that (c(L(D)), a)etf (M),o(F\ 2), End(L(D)) ® JC). If the condition ^jp- G Ou^Ui) is satisfied, then to calculate the first Chern class of L(D) one merely calculates the trace of the endomorphism given by a on UQ, this is exactly V(/o) fo • T h e o r e m 12 In the degree one graded piece of the cohomology ring o / M , o ( P , 2) we n 0 have an element d, which corresponds to the set of all points (C, f) such that C is reducible. In the cohomology ring d = s — 4b\. 2 Proof If we consider the divisor corresponding to the points (C, f) such that C is degenerate, on UQ it is given by the equation /o = m = 0, on U\ it is given by the equation / i = m! = 0. Since V(m) = m(s -Ab ), and V(m') = -m'(s' +s' b' +2b ), we see that ^ > 2 l 3 2 1 2 G 0 (Ui). Vi The corresponding element in the cohomology ring is s — Abi. • 2 T h e o r e m 13 The divisor of points (C,f) such that f(C) meets the linear subspace of P where a: _i = x = 0 is given by —s in the cohomology ring of Mo,o(P , 2). 71 n n n 2 Proof We consider first UQ. If x is such that C x is irreducible and f(C ) x required linear subspace, we have that w w „ _ i , i s + 6 _ist + r _ i t 2 n n = 2 ) 2 mr is + b st + r , t 2 n> 2 n 149 n 2 0 = 0 passes through the Chapter 4. Geometric Interpretation have a common solution. Thus fo = ( & n » n - l , l - & n - l » n l ) ( & n ' n - l 2 ~ K^n?) In the case C x + ^(^n,l n-l,2 ~ r r ) I n > 2 r _i,i) = 0. 2 n reducible the same equation holds. We calculate V(h) . —}— ^ , = - ( r i + r ) = -s . 2 Jo 22 2 On Ui we require that m'r' _ s + b' _ t + r' _ t 2 n 2 hl n mr' : s J - n lt2 = 0 = 0 + b' st + r' t 2 2 nA h = (Kr'n-1,1 lS n nt2 d < i ) ( & n < - i , 2 - C i < , ) + m(r' r' _ njl 2 n - r lt2 r - i , i ) = 0. 2 n > 2 n Again we calculate ^ = r 3 ! l + r , + 6' (r 3 2 1 2 i l +r 2 ) 2 )-26 . 2 The associated element in the cohomology ring is —s as required. 2 4.4 • Chern Classes and the Cohomology of M ,o(P ? 2) n 0 We have d = s — Ab\ ci = -s 2 2 + 61 1 c 3 = 2 1 61 ( - c 2 ,2 - 6xci - bf) . 150 Chapter 4. Geometric Interpretation thus we can write the generators 61, g and s in terms of d, C\, and c as follows: 2 s 3 = 2 g • + ci d 61 = 2 - d + 4ci _ 4 C - 2 d y 4c _ l . 2 + T h e o r e m 14 Let C\ and c be the first and second Chern classes of the vector bundle 2 E = 7r*/*0(l) on A^o,o(P ) 2). n reducible. Let n>2, Let d be the divisor of points ( C , / ) such that C is then the cohomology ring o/M ,o(P",2) is 0 C[d, ,c ]/I Cl 2 where I — (<7n+l, /n+15 1'n+l) and d-2ci 6 ( g +i ^ n d+4ci 3 . 4.5 4.5.1 , d +16c?+8dci g 2 -4c + /n+1 \ \ 2 ^n+1 / d+ci 3 V , / V d(^) 2 / Geometric Interpretation in the case n = 2 M , o ( P , 2) as the Blow-Up of the Veronese Surface in P 2 5 0 The image of the projective embedding i:P 2 P i:<a,b,c> < 5 a ,b ,c ,2ab,2ac,2bc> 2 2 2 is called the Veronese Surface. One can realize the space of complete conies (which is the classical equivalent of the coarse moduli space M o(P ,2)) as the blow-up of P 2 0] 151 5 Chapter 4. Geometric Interpretation along the Veronese Surface. We shall construct a mapping from our neighborhood UQ to this blow-up. We identify a point in P with a conic in P in the normal way, i.e. if a =< x , •.. , x >G 5 2 0 5 P then as long as x g" i(P ), x corresponds to the (possibly degenerate) conic with equa5 2 tion xX 2 0 + x Xl + x X\ x 2 + + xXX X XQX Z X 4 0 2 + xXX 5 x = 0. 2 This conic has the property that the variety associated to it is not a line (although it may be a pair of distinct lines crossing transversally). Thus there is a one to one correspondence between such points in P and points (C, /) in Mo,o(P ,2) where / ( C ) 5 2 is a non-degenerate conic or a pair of distinct lines (none of these points are stack points). Theorem 15 The coarse moduli space M ,o(P ,2) is isomorphic to the blowup of P 2 0 along the Veronese, P . 5 Proof We first analyze the blow-up in the neighborhood where x ^ 0, U , with coordinates 2 2 jj£ = uii for % — 0 , . . . , 5, i ^ 2. This neighborhood meets the Veronese at the images of points of the form < a, 6,1 > in P . Thus in this neighborhood the Veronese is described 2 by the following ideal: h = (w\ - 4w , w\ - 4twi, w w - 2w ) = (/ , / , i , 0 4 5 3 2)0 2 fo). When we blow-up along the variety given by this ideal we have the following Ui = {(x,y) G U x P | y =< y , y 2 2 0 y >, and yjf ,i( ) x u 2 for % < j , i,j = 0,1,2}. 152 2 = VihM 5 Chapter 4. Geometric Interpretation The neighborhood L is covered by three affines B i for i = 0,1,2, where r 2 2> B = 2>i {(x,y)eU \y ^0}. 2 i In particular B ,i can be described as follows: Since yi ^ 0, we have 2 Vo 72,0 = —/2,i -%(w -4wi) ™o = — - 72,2 = —/2,1 W = + wl 2 f 5 A . 3 Vi 2 so £?2,i can be identified with A with coordinates 5 2/o 2/2 Vi 2/i Since the point (m, 61, b , r ,i, r ) corresponds to the conic 2 2 2)2 ((62 - r ,i&i)X + ^2,1^1 - X ) {(b - r , h)X 2 0 2 2 2 2 0 + r Xx - X ) + m(r 2tl 2 2>1 - r ) X 2 2 > 2 2 0 =0 we have the following map from C7 to B y. 0 2 (m, by, bi, r ,i, r ) -»• ( r 2 2)2 2>1 r , - 2 6 + (r 2i2 2 + r )6i, - ( r , i + r 2)1 2)2 2 2>2 ) , 6? - 4m, -61), since 2/o ^0 = 2/1 = w\ - 4w «>5 /2,1 _ 4 w (r ,i -r , ) (6? - 4 m ) 2 0 = 2 2 l 2 ( 2,l - r 2,2) r 2 and m j%2 ^4^5 ~ 2^3 = = 2/i /2,i = -fel(?~2,l - r w -4wx (r ,i-r 2 2 2 | 2 2 ) 2 ) ) 2 2 If we take the subring of C[m, 6 , b , r i , r ] which is invariant under the action of Z , X 2 2) 2)2 2 it is generated by m, bx,b ,s , and 2 2 (r ,i ~ r 4 2 2 2 = ) 2 | 2 2 • So in "invariant" coordinates the map is as follows (m, 61, b , s , z ) -> (z + 2 2 2 2 - 2 6 + s b , - s , b\ - 4m, -b^j , 2 153 2 x 2 Chapter 4. Geometric Interpretation which is an isomorphism with inverse 2/o m\ [ v Ti 2 / 1 2 / 1 / {m) + 2/2 4 + w^ -Wi yx w 2 l 4 Let Ui be any other member of our etale open cover U for Mo,o(P ,2). Then since the 2 universal map for Ui differs from the universal map for UQ only by a projective change of coordinates for P , by choosing new coordinates for P we can construct an isomorphism 2 2 from Uil%i to an open neighborhood of P in the same way as above. Therefore since 5 M , o ( P , 2 ) is covered by the family {£/i/Z }, we have M o ( P , 2 ) = P^ as required. 2 2 0 4.5.2 2 Cohomology of M , o ( P • 2) v i a the Veronese 2 0 0i 5 The cohomology of the blow-up of the Veronese is easy to calculate, since the Chow Ring coincides with the cohomology in this case and the Chow Ring of a blow-up is readily calculable. Theorem 16 let X be a regularly embedded subscheme of a scheme Y, and let Y be the blowup of Y along X, X A Y X A Y if the restriction map i* : A*Y -> A*X is surjective then A*(Y) is generated by the class of the exceptional divisor as an A*(Y) - algebra. The relations are as follows: If we let T = j*[X], then aT = 0 */ aeA*(Y) and i*(a) = 0, r-l T + J2j T - + (-lY(i*[X}) = 0, r r k k 154 Chapter 4. Geometric Interpretation where r is the codimension of X in Y and^k are any classes of A*(Y) such that i*(7fe) = (-i) [X]r)c (N ). k k x/Y Proof see [7]. • In order to write down the cohomology ring we need to calculate the Chern classes of the normal bundle of the Veronese Surface in P . To calculate these Chern classes 5 we shall use standard properties of Chern classes (for example see Hartshorne, [11, p 429-431]). Lemma 42 The Chern classes of the normal bundle of the Veronese Surface in P are 5 ci(N) = 9h c (N) = 30h 2 2 where h is the hyperplane class in P . 2 Proof Since the Veronese is a non-singular irreducible closed subscheme in P , we have an exact 5 sequence 0 ->• T 2 p -> z*Tp5 iV -> 0 where N is the normal bundle of the Veronese in P and i is the Veronese embedding. 5 Therefore c (T 2)ct(N) = c (i*T 5). t t p (4.6) p where c (F) is the Chern polynomial of the locally free sheaf T. The exact sequence t 0 O 2 -> O p ( l ) ->> T 3 p 2 155 p 2 0, Chapter 4. Geometric Interpretation where the superscript 3, means the direct sum three times, gives c ( O ) ( T p ) = (Op2(l) ). 3 t p 2 C ( 2 Ct Now c (Op2) = 1 and C t ( 0 p ( l ) ) = (c (0p (l))) = (1 + ht) where h is the hyperplane 3 4 3 2 t 3 2 class in P (and h = 0). Thus 2 3 Q ( T 2 ) = 1 + Sht + ShH . (4.7) 2 P To calculate Ct(i*Tps) we observe that Ci(i*T ) = i*Ci{T s) = pS + Htf p where H is the hyperplane class in P . Now i*H = 2h since i i s a degree 2 map. Thus 5 (i* T p ) = (1 + 2htf = 1 + \2ht + 60h t 2 Ct (4.8) 2 5 since h = 0. We substitute 4.7 and 4.8 into 4.6, and letting Ci(N) be the i 3 th of the normal bundle JV, we have (1 + Ci(N)t + c (N)t ) (1 + 3ht + 3h t ) = 1 + 12/rt + 60^ t 2 2 2 2 2 thus 3h + (N) Cl =>ci{N) 3h + 27h + c (N) 2 2 2 ^c (N) 2 • 156 = 12h = 9h = 60h = 30h . 2 2 2 Chern class Chapter 4. Geometric Interpretation Lemma 43 The cohomology ring of the blow-up of the Veronese Surface in P is 6 C[T, H]/(TH ,T 3 - (9/2)T H + (15/2)T# - AH ), 3 2 2 3 where T is the class of the exceptional divisor, and H is the hyperplane class in P . 5 Proof Let h be the hyperplane class in P . Certainly the restriction map 2 i* : A*P -> A*P 5 2 H ->• 2h is surjective so A*(P ) is generated by T as an A*(P )—algebra. Thus is generated by T 5 5 and H as a polynomial ring over C. For the relations (-l) ^ 1 i* i = 7 r 2 = ~ H = *^ =^ = (-i) soh 2 7 -Yi 2 = ±H. 1 Thus the Chow ring has two generators H and T and the second relation is T 3 _ l2 T R + y* 2 T H 2 _ 4 i f 3 = Q 2 The last coefficient comes from the fact that U P ] = [K(P ) : 2 2 K(i(P ))]J^) 2 where [if(P ) : K(i(P) )] is the degree of the field extension (finite since i is an isomor2 2 phism) induced by the map i. The degree of this field extension is 2, and i(P ) = 2H . 2 3 For the first relation we look for elements a of A*(P ) such that i*(a) — 0, now the 5 157 Chapter 4. Geometric Interpretation generators of A*(P ) are of the form 7T and i*(7T) = (2/i)* which is zero exactly when 5 i > 3. Thus in particular i*(H ) = 0. Hence the first relation is TH 3 = 0. The Chow 3 ring is C[T, H}/(TH\T 3 - (9/2)T # + (15/2)T# 2 2 AH ). 3 • 4.5.3 Geometric Interpretation for n = 2 T h e o r e m 17 In terms of s and b\, T = —2(s — h) o,nd H — —s . Geometrically H 2 2 corresponds to the divisor of points (C,f) 2 such that f(C) contains the point < 1,0,0 > in P , and T corresponds to the stack locus. 2 Proof The result for H is just a special case of Theorem 13 The stack locus (which corresponds to the exceptional divisor in the blowup) is given by Z2 = 0 on UQ- Thus corresponds to the element in the cohomology ring as long as z divides V(z ). Now 2 2 V(z ) = -\d V(d2) 2 = 2 - s ) = 2z2(h 2 s) So the corresponding element in the cohomology ring is —2(s — &i). 2 Suppose we take T = - 2 ( s 2 - bi) and H = -s 2 r = b\ (s - 46i) 2 158 • we get the same relations as in the blow-up case. In our original calculation we had relations x 2 Chapter 4. Geometric Interpretation r = (s - 26i)(s| - s 6i + 46 ). 2 2 2 2 Repeated division yields r = - ( - 8 6 ? + 6s 6 - 36is + sij) = 2 2 2 9 2 15 (-232 + 260 - - ( - 2 3 + 2 6 ) ( - s ) + — (-232 + 3 2 1 2 = r 3 _ 9 2 r 2 / / 26 )(-3 ) 1 2 - 4(-s ) 3 2 2 }$ 2_ 3 + TH 4H and - 8 r i + (2s + 46i)r = -86?(s - 46 ) + (2s + 46i)(-86 + 6s 6 - 3&i3 + s ) 3 2 2 2 x 2 2 2 3 2 = ( - s ) ( - 2 s + 26 ) = H T 3 3 2 2 1 Therefore our cohomology calculation is consistent with the blow-up calculation. 4.6 Chern Roots and the cohomology of Mo,o(P , 2) n In this section we give the cohomology ring of M o ( P , 2) in terms of the Chern roots of n 0) the vector bundle E. Theorem 18 Let 7 l ; 7 and 73 6e the roots of the Chern polynomial C(t) of E, where 2 C{t) = r + c t + c t + c 3 2 x 2 3 for Ci the Chern classes of E, and 71 = 61 in the notation of before. Let Z ring C [ i , 72,73] as follows: 7 71 7i 72 ->• 73 73 ->• 72 2 act on the Chapter 4. Geometric Interpretation Then the cohomologyringo/M o(P",2) is the invariant subring ofthe quotient ring 0) C[ i,72,73]/(#i,#2,#3) 7 where the Ri are invariant polynomials and are given by 7" Ri R2 2 +1 ~ 7? 7? + 72 - 7 i +(7 72-7i n + l 2 + 1 ~ 7i" +1 73 - 7 i _(2£^) = \ Rs + 1 ( 7 3 _ _(23^r, 7 l ) / V +7r )+(7 n + l i 3 ( 7 2 _ 7 l ) 73 - 7i +7r ) i 7r ((72-7i) + (73-7i)) = + 1 Proof Let u = s - 26i and b = b . Then the cohomology relations for M , o ( P , 2) are given by n 2 x 0 the matrix ( 2" -uu 2s 0 \ \g \u -2 V0 0 b J u + 2b \ 1 and vector V - p + 46 2 \ b {u - 26) J 2 i.e. the cohomology ring of M ,o(P", 2) is 0 C[b,u,g]/I where / = (s , f , n+1 n+l r ) n+1 and ( S +1 n ^ A ~ v. n fn+l 160 l Chapter 4. Geometric Interpretation We shall show that the eigenvalues of A are exactly the Chern roots, and we shall write A in terms of the Chern roots and diagonalize. We then calculate the A ~ using the n l diagonalization and the relations will follow. The eigenvalues of A are the roots of (b-\)\[\u-\) -\g\=V thus We write the Chern polynomial in terms of b, u and g. Recall that Ci = bi — s = —u — b ° = -\g + \sl-b = h 2 c 3 2 = ~g + ^(u + 2b) -b 2 2 - ^s + s h -b ^=b 2 - 2 2 2 = ~g + ^u + ub 2 ^ Thus C(t) = t + (-« - b)t + {~\g + \u + ub)t + h(jg - ^u ) 3 2 2 ( * - * ) ( ( | « - t ) 2 a - | / so the Chern roots are the same as the eigenvalues. Suppose we take 7i = b u + y/g 72 2 u-^/g 73 2 ' u = 72 + 73 9 (72-7s) - = 2 Thus C[b,u,g] is isomorphic to the invariant subring of €[71,72,73] under the Z action 2 which sends 72 to 73, 73 to 72 andfixes71. 161 Chapter 4. Geometric Interpretation In terms of the Chern roots, the matrix A is \ (72 + 7s) |(72 - 7 s ) V 2 \ 0 5(72 + 73) -2 0 7i J 0 We now need to find eigenvectors of A corresponding to the eigenvalues, 71, 72 and Let / 1 -(72-71) - ^ ,v = (73-71) -(72-71X73-71) ( (72 - 73) 2 / ) 1 \ 0 ,v = 3 J ( I 1 then Vi, V and V are eigenvectors and if 2 3 \ E vi v v 2 ; Then A n = ED n E l 3 ; ; / , where D is the diagonal matrix of eigenvalues. Now 0 E -1 1 2 0 (73-71) (72-71) 1 1 (72-73X72-71) 2(72-73 V I 1 i \ 2 2(72-73) ± / (72-73X73-71) and A ~V n l = ED - E~ V n \ l as required. • 162 l R2 / ^ -(72 - 73) 0 / Bibliography [1] E . Akildiz and J.B. Carrell. Cohomology of projective varieties with reglular SL actions. Manuscripta Math., pages 473-486, 1987. 2 [2] K . Behrend, D. Edidin, B. Fantechi, W. Fulton, L . Gottsche, and A . Kresh. Introduction to Stacks, in preparation. An [3] K . Behrend and Yu. Manin. Stacks of stable maps and Gromov-Witten invariants. alg-geom/9506203. [4] A. Bialnicki-Birula. Some theroems on actions of algebraic groups. 98:480-497, 1973. Ann. Math., [5] J.B. Carrell. Vector fields, flag varieties and the Schubert calculus. Proceedings of the Hyderabad Conference on Algebraic Groups, pages 23-57, 1989. [6] J.B. Carrell and D.I. Lieberman. Vector fields, Chern classes, and cohomology. Proc. Sympos. Pure Math., Vol. XXX:251-254, 1975. [7] B. Fantechi and L . Gottsche. The cohomology ring of the Hilbert scheme of 3 points on a smooth projective variety. J. Reine Angew. Math., 439:147-158, 1993. [8] W. Fulton and R. Pandharipande. Notes on stable maps and quantum cohomology. alg-geom/9608011. [9] E . Getzler and R. Pandharipande. The Poincare Polynomial of Kontsevich's space of stable maps M.o (p , d). in preparation. n >n [10] T . Graber and R. Pandharipande. Localization of virtual classes, alg-geom/9708011. [11] R. Hartshorne. Algebraic Geometry. Springer-Verlag, 1997. [12] M . Kontsevich. Enumeration of rational curves via torus actions, hep-th/9405035. [13] H. Matsumura. Commutative Ring Theory. Cambridge University Press, 1992. [14] R. Pandharipande. Intersections of Q-divisors on Kontsevich's moduli space Mo, (p ,d) and enumerative geometry, alg-geom/9504004r n [15] J.P. Serre. Geometrie algebrique et geometrie analytique. Ann. Inst. Fourier, pages 1-42, 1955-1956. 163
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The cohomology of Kontesevich’s stack of stable maps to Pⁿ, the case of conics O’Halloran, Anne Fionnuala 2000
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Title | The cohomology of Kontesevich’s stack of stable maps to Pⁿ, the case of conics |
Creator |
O’Halloran, Anne Fionnuala |
Date Issued | 2000 |
Description | In this thesis we consider the singular cohomology of M₀,₀(Pⁿ,2), the coarse moduli space associated to Kontsevich's stack of degree two stable maps to Pⁿ, M₀,₀(Pⁿ,2). We show that the cohomology ring is generated by a divisor d which corresponds to the locus of pairs (C,g) with C reducible, and the first and second Chern classes, c₁ and c₂, of the canonical rank three vector bundle E = π۰f0(1) on M₀,₀(Pⁿ,2), where π is the canonical projection associated to the universal curve C and f is the universal map. We give the cohomology ring as a quotient of a polynomial ring in these generators. The relations are in degrees n, n + 1 and n + 2. We also give a representation of the cohomology ring in terms of the Chern roots of E. The results are conjectural for n >> 0. |
Extent | 5012796 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-09-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080018 |
URI | http://hdl.handle.net/2429/13179 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2000-11 |
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UBCV |
Scholarly Level | Graduate |
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