T and ip is a G-equivariant S-morphism s*E \u2014> s*X. This definition generalizes to the case were E \u2014> T and X \u2014> T are representable morphisms of algebraic stacks. Then Mov^(E, X) is an algebraic stack over T with the property that the structure morphism MovG (E, X).\u2014> T is representable. Let X be a G-variety over C . We want to construct a space whose points are G -equivariant morphisms from G-curves into X. Consider the diagram arc? where

U { o o } is transitive. Let T(2) be the principal congruence subgroup of level 2, which is defined as the kernel 1 -> r(2) PSL(2 ,Z) -> PSL(2 ,Z \/2Z) -> 1 of the morphism reducing the matrix coefficient modulo 2. The reason we consider this group is that the quotient r (2) \\H is conformally isomorphic to P 1 \\ {0,1, o o } , and the quo-Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 27 tient morphism H \u2014> P 1 \\ {0,1, oo} is a universal cover. It extends to a unique continuous T(2)-invariant morphism u : H \u2014-> P 1 . Consider the 3 orbits of F(2) on Q U {oo}: { GVG7L 1 \u2014 - I odd G 2Z + 1, even G 2Z } odd J : = \/ ^ i , o d d . \u20ac 2 Z + 1 (oaa2 00 odd even odd G 2Z + 1, even G 2Z \\ {0} > U {oo}. Hence after composing with an appropriate automorphism of P 1 \\ {0,1, oo} we can assume that u :. . P1 I H I oo H-\u2022 oo. For each i G {0,1, oo} denote by Fj(2) the stabilizer of i in T(2). These are cyclic groups generated by the parabolic transformations OJQ 1 0 - 2 1 e r 0(2), a i 1 - 2 2 - 3 G ri(2), a o o : = 1 2 0 1 e rQ O(2). These are chosen so that OOOIOQO = id. The group F(2) is a free group generated by any two elements of { 0 0 , 0 : 1 , 0 : 0 0 } . The automorphisms 0 0 , 0 1 and Ooo \u00a3 T(2) are called the monodromies of respectively 0, 1 and 0 0 . This terminology is justified by the following construction. Let G be a finite group, and let m\u00ab G G for i G { 0 , 1 , 0 0 } be such that mominioo = id. We will construct a canonical object e m of M3'm. Let 6m : T(2) \u2014> G be the homomor-phism given by #m(oj) = mj for i G { 0 , 1 , 0 0 } . Denote by Gm the right F(2)-space G with structure given by 9m. The group T(2) acts on Gm x H b y a - ( g , i ) = {g6m(a)~l,a \u2022 x), and let Em = Gm *r(2) ^ be the quotient by this action. Denote by [g, r] the im-age of the pair (g,r) G Gm x HI in Em, and let u m : EI \u2014> Em be the composition -\u00bb Gm x 3HI, x '\u2014\u2022 (id,x) with this quotient. For i G { 0 , 1 , 0 0 } , let of the inclusion : um(i) = [id, i] \u20ac Em, and let \u00a3\u201e Gn cr(2) Lemma 7.1. The triple em := ( E m , S m , < T m ) is an object ofM3'm. Proof. We need to verify the conditions in the Definition 2.4 in Chapter 2. Condition (a) is trivial as Em is smooth. Condition (b) follows from the fact that T(2) acts freely on H and that the union of the three orbits OQ, 0\\ and Coo is the complement of EI in HI. For (c) we need the quotient of Em by G to be P 1 which follows from the fact that the quotient of H by the action of T(2) is P 1 . That E m i is a G-orbit of ami follows from the fact that Oi is a T(2)-orbit. Hence we only need to show that the monodromy of am^ in Em is mj = #m(oj) for each i G { 0 , 1 , 0 0 } . Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 28 We start with i = oo. Consider the open set H i (see Section 7.1). It is an Qoo-invariant neighborhood of oo. Moreover ao \u2022 H i n H i = 0. It follows that the image of H i in Em is isomorphic to the quotient ( a ^ m o o ) ) \\ H i . But ( a t m i n f t y ) ) \\ H i is r o o(2)-equivariently isomorphic to an open disk on which GJoo acts by multiplication by exp (^ol?\u2122 ))\u2022 This isomorphism is given by r i\u2014> exp ^ o ^ T for r ^ oo and oo i-> 0. The result follows. The case i = 0 (resp. i = 1) follows from the case i = oo by applying the automorphism z i * ~~ (resp. z i\u2014> to H . It sends 0 (resp. 1) to oo and conjugates ao (resp. a\\) into a o o . \u2022 Q 7.2 Canonical presentation of M3 Q Recall that the monodromies at the markings give rise to 3 maps fii : A43 \u2014> G, for i G {0,1, oo}. Translation of the markings gives rise to 3 actions of G on M3 , and since the monodromy of a marking stabilizes it, by Lemma 6.2 we have 3 actions of the groupoid G( G ) on M3 over G (where the anchor \u2014> G for each action is the monodromy morphism \/ i , corresponding to the marking being translated). Let G2G^ act on A43 over G2 by translation of the markings indexed by 0 and 1. Consider the morphism G2 \u2014\u00bb , given by (mo, mi) i\u2014> e m where moo = ( m o n i i ) - 1 . For simplicity, denote the element ([go, mo], [gi, mi]) of G?Gs by [g, m]. We have a morphism :G2{G)->M\u00b0 [g,m] H+ C [ g i m ] :=g- en where g \u2022 e m is the translation of the marking 0 by go and the marking 1 by g i . We will show that e_ : G2G^ \u2014> is a stack quotient by a right action of the group T(G) (G) over G. The anchor for this action is given by G ? G 1 \u2014> G , [g, m] i\u2014> momi, and the action is given by ([go, mo], [gi,mi]) -a = ( [g 0 a,a _ 1 m 0 a] , [gia, a _ 1 mia]) for a G (moo)- This action preserves the anchor map, and hence is an action of a group over G . Now let c a : C[g j m] \u2014> \u00ab[g,m]-a D e the arrow of M3 given by the G-equivariant isomorphism of G-curves EH*M = \u00b0 m xr(2) H - G a ~ l m a x r ( 2 ) H = \u00a3 C [ g m ] . a [h,r] i-> [ha,r]. Note that this is a well defined isomorphism as # a - i m a = a - 1 # m a . P r o p o s i t i o n 7.2. The functor e_ is an isomorphism of groupoids Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 29 Proof. First we prove the essential surjectivity. Let e be an object of M3 . Let E%\u00b0 be the connected component ofP 1 \\ {0,1, oo}. After acting by an automorphism of HI we can assume that uc(oo) = a e ) O 0 . Let rm. be the monodromy of ut(i) for i \u00a3 {0,1, oo}. For i \u00a3 {0,1}, there is a unique g, \u00a3 Gj ( ir i j ) such that gi \u2022 Uc(i) \u2014 (Te,i-We have an element ([go, mo], [gi, mi]) of G2Gy I claim that e[gi in] c\u00b1 e. The morphism ut : H -> Et induces ve : Eg-img := Gm x r ( 2 ) H -> i? e given by v c([g,r]) = g \u2022 ut(r). This is injective as F(2) acts transitively on the intersection of any G-orbit on Et with the connected component E%\u00b0. It is surjective as any point of Et can be translated by an element of G into E%\u00b0. Moreover ve(o-t[gm,ti) = ve([gi,i}) = gi-ue(i) = ot^ for i \u00a3 {0, l ,oo}. This shows that ve is the required isomorphism. We now prove the full faithfulness. Let ip be an isomorphism between ei := e[g,m] and e 2 : = e[h,n]- By definition we have maps u, : HI \u2014> EH such that p e , i \u00b0 \u00ab i = Pe,2 \u00b0 \u00ab2 where p c,j is the quotient morphism Etj \u2014> P 1 . As ^ is a G-isomorphism \u00a3?C l \u2014> i? C 2 over P 1 , there exits a unique (and hence we have the faithfulness) a \u00a3 G such that IP(U\\(T)) = a \u2022 V,2(T) for all r \u00a3 H . The morphism ?\/> sends markings onto markings: i>(o~ei\/) = cre2}i. For i = oo we have ^(^(oo)) = a-U2(oo) = 1*2(00), which implies that moo = rioo and that a \u00a3 (rioo)-For i \u00a3 {0,1} we have ^(gi \u2022 \" i W) = g\u00ab \u2022 W) = g i a \u2022 u 2 ( \u00ab ) = hj \u2022 u 2 ( i ) and it follows that gjmjgr 1 = hmjh^ 1 and h~ 1gja \u00a3 (n,), which shows [g, m] \u2022 a = [h, n]. Hence the full faithfulness. \u2022 7.3 Compactification of the moduli space of G-equivariant morphisms from a G-cover of P 1 with 3 ramifications Denote by J7^ the subset of G 3 of triples (mo,mi, moo) such that momimoo = id. Given m \u00a3 A \/ - 0 , we consider the scheme of G-equivariant morphism M3G'm(X) := MorG(Em,X) from Em to X, and the subschemes Af^'m(X,P) of morphism of a given degree 0 \u00a3 A+(X)G. We are looking for a compactification of M^'m(X, 0) such that the 3 eval-uation maps extends. Notice that N^,m(X, 0) is the fiber of the forgetful morphism M3'm(X, 0) \u2014> 97t G , m above e m . Hence the fiber above e m of the composition M^'m(X, 0) -9JtG'm \u2014s- A ^ ' m (where the second morphism is the stabilization morphism) is proper and contains A ^ ? , m ( X , 0) as a dense open substack. The evaluation morphisms extend to the morphisms induced by the evaluation morphisms of M3 'm{X,0), and hence it is the desired compactification. Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 30 Def in i t i on 7.3. Define 77G(X,P) (resp. [X, \/?)), by the cartesian diagram 77% (x, p) \u2014 - M33'G(x, p) \u2014> M\u00b0(X, p) 77\u00b0 \u00bb- G \\ G ) MG where 773 -* G2G) is given by (m 0 , m i , m ^ ) ^ ([id, m 0 ] , [id, mi]). Define the virtual fundamental classes [77\u00b0(X, p)]vir (resp. \\M3\u00b0{X, P)]vir) of 77\u00b0(X, p) (resp. Ai$ (X,\/3)) by pulling back the virtual fundamental class of A 4 3 (X,f3). L e m m a 7.4. Consider the left G2G^ action on M3 (X, p) by translation of the two first sections. Then the composition 77^ (X, P) \u2014\u2022 M3 (X,P) \u2014> [G2G G2G^ where the second map is the \/~<2 , s T(G) ~^ L \u00b0 ( G ) \\^(GX product map of the groupoid. It follows that the composition G2 \u2014> G2G^ \u2014> [G2G^\\G2Gy is an isomorphism, and the result follows from the definitions. \u2022 7.4 Inflation Suppose \/ f is a subgroup of G. To distinguish between the groups H and G, let tH and e G denote the morphism e of the last section corresponding to the given group. For m \u20ac. 77 3 , the inflation of is naturally isomorphic to e G . It follows that the inclusion of H into G induces inclusions M3 \u2014> . M 3 ' and Tv^ -> Tv^ such that the diagram 77\" - T v f M3 M \u20223,G H I < ; I X 7 G 7VI3 ^ A I 3 is cartesian. We pullback the definition of inflation to stack of if-maps. Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 31 Lemma 7.5. We have a commutative diagram 77\u00b0 (X) M%(X) A 7 f ( x ) where all faces are cartesian. Let evg :77%(X) -+ (X< G ) ) 3 be the composition of 77%(X) -\u00bb J^(X) with the product of the three evaluation maps evj : 7*4%(X) \u2014> X^G\\ P r o p o s i t i o n 7.6. Given a \u20ac A + ( X ) G , we have a cartesian diagram UP^f(x,p)l^77%(x,a) (x^y where the disjoint union is over all 0 e A\u00b1(X)H such that a = $ 2 g e G \/ H * s cartesian. Proof. The result follows by cartesian diagram chasing from Lemma 7.5, Corollary 5.13 and the following two cartesian diagrams ( x w ) 3 \u2014 - (x^y H3 ^ G 3 and H3 ^ G 3 \u2022 32 Chapter 8 Group quantum cohomology In this section we first review the definition of group cohomology of [FG03]. Then we define the algebra of invariant quantum parameters, and build the group quantum cohomology as the tensor product of this algebra with the group cohomology. We define a product on the group quantum cohomology using G W type integrals over the moduli stack constructed in Chapter 7. We then prove a functoriality property of the group quantum cohomology under the inclusion of groups with compatible actions on the variety X. 8.1 Group cohomology The following is an overview of [FG03]. Let G be a finite group acting on a smooth projective variety X. We first define the \"cohomology\" group. Definition 8.1. Using the intersection theory with coefficients in Q , define the Q-module A*{X,G) :=A*(X^). It is a left G-module under the action given by g - a ^ g - 1 ) * * for g G G and a \u20ac A*{X^). Let A*(X,G)m be the direct summand of A*(X,G) cor-responding to the closed and open subvariety {m} x Xm C X^G\\ Give A*(X,G)m a Q-grading as follows: given c G Q , let Ac(X,G)m := 0 Ac~ : H \u2014> G is a homomorphism of groups. Then 4> induces an action of H on X. Define A*(X,4>) : A*(X,H) A*{X,G) as the morphism of iJ-modules induced by pushing forward under the open and closed Jf-equivariant morphism X ^ \u2014> X^G\\ Proposition 8.3. This defines a functor A* (X, \u2014) from from the category of groups acting on X and homomorphisms compatible with the actions to the category of Q-graded Q-modules. Chapter 8. Group quantum cohomology 33 8.2 Quantum parameters Let G be a finite group acting on a smooth projective variety X. Definition 8.4. Denote by Q*(X,G) the semi-group algebra of the semi-group of G -invariant effective curve classes Af(X)G, that is the algebra on the symbols qG with P e A+(X)G, and relations qG\\^ = qG1+p2 for & e A+(X)G. Denote Q*(X,G) by Q*(X) when G is trivial, and its generators by q13 with P e A^(X). The algebra Q*(X, G) is Q-graded by degqG:=j^P(ClTx). where T\\ is the tangent sheaf of X. Suppose H is another finite group, and : H \u2014> G is a homomorphism of groups. Then (p induces an action of H on X. Define Q*(X, = ^ G ' * \u2122 g ' P which induces a homomorphism of algebras Q*(X, G, the homomorphisms Q*(X,

) and A*(X,4>) induce a homomorphism of Q-graded Q[if]-modules qA* (X, ) : qA* (X, H) \u2014> q A * ( X , G) . Let i be the involution of X^ sending the component {g} x Xs isomorphically onto { g - 1 } x X&~\\ using the identification X% = X^'1. Consider the diagram ^(x , \/ j )^ (x^) { 0 - w \" G , i Chapter 8. Group quantum cohomology 34 where TTG,I is the projection on the ith factor for i G {0,1, oo}. Given pi G A*(X,G) for i G {0,1}, define P0*GPl- = qG(L\u00b0 ^G.oo)* (^GflPO \u2022 7I\"G,lPl 1 ( e V o ) * J7^{X,0) M- \u00bb>g,g Lemma 8.7. If 0 ^= 0, then the morphism f forgetting the first marking is the universal G-curve over A4^'s'e (X,0). Proof. This result follows from a widely accepted result of Abramovich and Vistoli (it appears without proof.in [AV02]): the stack of twisted stable maps M2([X\/G], 0) (where [X\/G] is the quotient stack) has the universal curve identified with a stack that they denote by M~'3([X\/G],0). It turns out that this stack pulls back to A f G ' ' d ' g ' s (X,\/3) under the morphism M%'e's (X,(3) M2([X\/G], 0) that we will construct in Section 8.7. The result follows from the fact that the universal G-curve over Ai2([X\/G),0) pulls back to the universal G-curve over M2 .g.g (X,0). \u2022 P r o p o s i t i o n 8.8. This makes q A * ( X , G) into a Q-graded algebra with identity. Proof. First of all we prove that the product preserves the Q-grading. This follows from the computation of the dimension of the virtual fundamental class at f: d i m X + d e g \/ f * T X - a (m 0 , \/ f (c r f , 0 ) ,X) - a(mi, \/ f ( X% and 7T2 : X& x X&~ Xs 1 are the projections. First notice that J\\f^'ld's's (X, (3) \u2014 0. First A i f ' l d ' g ' g \\x,0). We will show that if 0 0, ev* [ . M G ' i d ' g , s \\x,f3) we note that the morphism ev factors through the morphism \/ :, A ( G ' l d ' g ' g (X, (3) \u2014* Q \u201e \u201e \u2014 1 A42' ' (X, 0) forgetting the first marking. As this morphism is proper of pure codimen-sion 1 by Lemma 8.7, we have \/ , 0. J^M^\\x,0) 0 which implies that ev* A l G ' l d , s ' s \\x,0) 7<3,id,g,g '3 the identity on A * ( X s ) . The proposition follows. If 0 = 0, then AT 3 , , u ' 6 ' 6 ( X , 0) = Xs and the above product is just the product by \u2022 Chapter 8. Group quantum cohomology 35 8.4 G-graded C7-algebra and the braid group action Definition 8.9. A G-algebra A is called G-graded if there is a decomposition A = 0 g 6 G A 6 compatible with the G-action in the sense that for all g and h s G 1. h \u2022 A g C A h g h _ 1 and 2. A6Ah C A g h . It is G-graded commutative if 0g<7h = (g \u2022 (momim^^mo .n ioo) hoc \u2022 (moj in^moo) H-> (rriojmimoomj^.mi) These two maps preserve 77% C G 3 and respect the braid relation, inducing an action of B3 on 77%. Lemma 8.10. The morphism e_ : 77% \u2014\u2022 7^C% is B3-equivariant. Proof. We will prove equivariance for the action of 601 > the case of &ioo being similar. Let m G 77% and n = 601 ' m - We need a G-equivariant isomorphism : Em ~ En such that {omt00) = a\u201e>00. Recall that Em := G m x F ( 2 ) H and En :\u2014 G n x F ( 2 ) H (see Section 7.1). The above conditions are equivalent to 0([id, 0]) = [id, 1], 0([mo, 1]) = ^>([id,Q!o \u2022 1]) = [id,0] and <\/>([id,oo]) = [id, 0 0 ] . Let 1 1 0 1 G PSL(2 ,Z) Chapter 8. Group quantum cohomology 36 and let G n x r ( 2 ) HI be the unique G-equivariant isomorphism such that 0([id,r]) = [id, 0O(T ) ] . That (p is well defined is a consequence of the relations (Poao4>o 1 = a i a n < i (t>oai4>Ql = a r 1 Q ; O Q : I - That it has the above properties is a consequence of the following calculation:

((momimQ 1 , m 0 \u2022 xi) , (m 0 , x0), (moo, \u00a300) ) &I00 : ((m 0,a;o), ( m i . n ) , ( m o o , ^ ) ) H-> ( ( m o , i 0 ) , ( m ^ o o m ^ m i \u2022 Xoo) , ( m i , i i ) ) making the evaluation ev : 77%(X) \u2014> ( x ^ ) ) * 0 ' 1 ' 0 0 * into a B 3-equivariant morphism. Consider the following G-actions on 77%, M% and 7sC%(X). Given g G G , and m G 77% let <5g-m = (gmog - 1 , gmig _ 1 , gmoog _ 1 ) . On 7*4% and more generally on 7^(%{X) the action is 5g \u2022 f = t 0(g) 0 *i(g) 0 *oo(g) \u2022 f for any object f of 74% or jrf%{X). Lemma 8.12. The morphism e : 77% \u2014> M% is G-equivariant under the above actions. Proof. Given g G G and m G 77%, we need a natural isomorphism 5g \u2022 c m ~ e Gs^m x r ( 2 ) H , [h, r] h-> [hg\" \\ r] . \u2022 As before, this induces a unique G-action on 77%(X) such that the morphisms to 77% and _\/Vf3 (X) are G-equivariant. The virtual fundamental class of 77%{X,0) is invariant under this G-action because the virtual fundamental class of 7^C%(X, 0) is invariant un-der any translation of the markings and because of the G-equivariance of the morphism 77%(x,0^7J%(x,p). Chapter 8. Group quantum cohomology 37 Theorem 8.13. The group quantum cohomology qA*(X,G) is a G-graded commutative G-algebra. Proof. To show that it is a G-algebra we need to show that the G-action is compatible with the product structure. Suppose : H \u2014> G of groups, the morphism qA* (X, cp) is a homomorphism of Q-graded algebras. Proof. We have to show that for pi \u20ac A*(X,H), qA*(X, )(pi) *G qA*(X,(j>)(p2). F ix a e A f ( X ) G . By Proposition 7.6 il [jvf (X,a) \/3\u20acA+(X)\"\\ Infg\/3=a A^\"{X,I3) where i : ( X ^ ) 3 -> ( x ( G ) ) 3 is the product of the inclusion j : X ^ -> X ( G ) . Since ev f f is proper and i is a regular embedding (ev#)*i ! = i ! (ev G )* (see [Ful98] and [Vis89]). Now Chapter 8. Group quantum cohomology 38 we calculate: qA*(X,cA)(pi *H p2) = = Yl

)\\(LonH,3)* (n*HtlPl-n*Hi2P2 \u2022 (ev*), [jvf (X, \/ ? ) ] } ( (i 0 7Tff,3)* \"\"H.IPI \u2022 7T\/f,2P2 \u2022 (ev#), ] T [X^f (X,\/?)] \/ 3 6 A + P O \" Infg fl=a \/ J = ^ <7GA*(X,0) { ( t o 7 r f f , 3 ) . ( T T ^ X P I \u2022 7r^i2p2 \u2022 (ev\u201e)\u00bbz ! [ A ^ X ^ ) ] \" \" ) j = \u00a3 QGJ*(,L \u00b0 ^ , 3 ) * f ^ i P i ' 7i#,2P2 \u2022 i ! ( e v G ) , [ A f ( X , a)] J \u00a3 9G(<- 0 7TG,3)*i*i* f TTG,IJ*PI \u2022 \u2022xG,2J*P2 \u2022 (evG)* [ A ? (X, a)] a 6 A + ( X ) G = \u00a3 0 ^ G ^ ) * f TT*G,I3*PI \u2022 nG:2J*P2 \u2022 (evG)* [77\u00b0(X, a)] J Q \u00a3 A + ( X ) G = qA* (X, 4>) (Pl) * G qA* ( X , 4>)(p2). \u2022 8.6 Degree 0 specialization and Fantechi-Gottsche group cohomology In this section we prove that the degree 0 specialization of the group quantum cohomology is isomorphic to the group cohomology of Fantechi-Gottsche (see [FG03]). Let m \u20ac 77\u00b0', and suppose we have cr, G A * ( X , G) supported on {m,} x X m * for i G {0,1}. We consider the coefficient of qG in the product ao * G o-\\. Notice that ATG'm(x,o)~xmonxm\\ and denote this intersection by Xm. Let TT : Em x Xm \u2014> X m be the universal curve over Xm, and let \/ : Em x Xm \u2014> X be the universal G-morphism . Then \/ is the unique G-equivariant morphism whose restriction \/ \u00b0 to Em x Xm (where Em is the connected component of Em containing all the markings crm >i) factors through the projection TT\u00b0 : Em x Xm -y X m and the natural inclusion tm : X\u2122 -> X. The obstruction theory on A \" G ' m ( X , 0 ) ~ Xm is RTTg (f*flx and its dual is RTTGJ*TX- The 0th cohomology of the above complex is just the tangent sheaf Tjfm of Chapter 8. Group quantum cohomology 39 Xm which is locally free as Xm is smooth. It follows that R7r G \/* is locally free, and by [BF97] [Proposition 5.6], the virtual fundamental class is given by '77%'m(x, o)]m r = cr R 1 ^ \/ * TX \u2022 [Xm] = ( - l ) r Cr 1TG (f*nX \u00ae LOVm) \u2022 [Xm] where r is the rank of R 1 ^ 0 \/ * T ^ - This is precisely the class defined in [FG03], and this proves the following result. Proposition 8.15. Consider the Q-graded ideal qA*(X,G)+ in qA*(X, G) generated by all qG with 0 G A~l(X)G such that 0^=0- Then there is an isomorphism qA*(X,G)\/qA*(X,G)+ ~ A*(X,G) of Q-graded G-rings, where A*(X, G) has a ring structure defined in [FG03]. Note that the fiber of R V G \/ * T X over a point x G Xm is B.Em(f* TX)G, where fx : Em \u2014> X is the unique G-equivariant morphism which is constant on with value x. The rank r of R 1 7 r G \/ * T x at x can be computed using the equivariant Riemann-Roch formula: r = d i m x Xm \u2014 d i m X + a(mi, x, X) + a(m2, x, X) + a(m3,x, X). 8.7 Orbifold quantum cohomology There is a natural inclusion of the orbifold quantum cohomology of the quotient stack into the group quantum cohomology. This inclusion turns out to be a homomorphism of Q-graded algebras, and it identifies the orbifold quantum cohomology with the G-invariant part of the group quantum cohomology. We review the definition of orbifold quantum cohomology (see [AV04]), which we will need only in the case of a global quotient stack. Definition 8.16. Suppose that a finite group G acts on a smooth projective variety X. Let A*orb([G\\X}) := A*( [G ( G ) \\A- ( G ) ] ) * A*([G\\X(G>]) * A*(X^f. Let $X,G be the quotient X^ -> [ G ( G ) \\ X ( G ) ] , and let * x , G = Krb([G\\X}) = A*( [G ( G ) \\X( G >] ) - A*(X^) = A * ( X , G ) be its pullback. There is a unique Q-grading on A*rb([G\\X}) making <&*x G into a morphism of Q-graded abelian groups (with the Q-grading on A*(X, G) given in Definition 8.1). Consider the action over G 2 of G2G^ on ~M%(X) by translation of the markings 0 and 1. Let \/C 3 (X) := [G2G^\\A43 (X)] be the quotient stack. It is a moduli stack whose objects are admissible balanced G-curves with 3 twisted sections where the section oo is trivialized. Hence it is isomorphic to the moduli stack of twisted stable maps into [ G \\ X ] , with a trivialization of the twisted section oo (see [AV02] and [AGV02]). It Chapter 8. Group quantum cohomology 40 has a universal G-curve and a universal G-morphism, and hence an obstruction theory and a virtual fundamental class. The morphism .A43'G(X) \u2014> M^(X) constructed in Definition 7.3 is GfiL equivariant and hence induces a morphism JC3'G(X) \u2014> K,G{X) on :3,G, the quotients, where K.3 (X) : \\G2G^\\M3 (X)}. We pull back the virtual fundamental class of lCx(X) to IC3'G(X). By Lemma 7.4, we have a morphism J7\u00b0(X) Ml'G(X) such that the composition with the quotient Ai3 (X) have the following commutative diagram \/C 3 ' (X) is an isomorphism. We 77\u00b0 (X) \u2022M33'G(X) )cl'G(x) \u2022M^(X) \u2022JC\u00b0(X) and the virtual fundamental classes are compatible under any morphism in the above diagram as the universal G-curves and G-maps pull back. In particular <\/>*[A\/\"G(X, f3)]mr = [icl'G(x,py as [ G ( G ) \\ X ( G ) ] 2 x X^ on the quotients. In particular the M33'G(X) ( l ( c ) ) 2 x l ( c ) >cl'G(x) [G(G)\\X{G)? x is cartesian, where the vertical arrows are quotients by (G( G ) ) 2 . In [AGV02], the G W invariants for the orbifold quantum cohomology are integrated Q over the stack K,3 (X), and then multiplied by the order of the monodromy of the third marking. By integrating over IC3 (X) we will obtain the same numbers as K.3 (X) \u2014> K.3 (X) is an etale morphism whose degree is locally equal to the order of the monodromy 3 Q Q of the third marking. This follows from the fact that M3 (X) \u2014> Ai3 (X) is a stack quotient by the right action of (G) (by Definition 7.3 and Proposition 7.2). Definition 8.18. The orbifold quantum cohomology of [ G \\ X ] is the Q-graded algebra qA*([G\\X]) := Q*(X,G) \u00aeQ A ^ ( [ G \\ X ] ) with the multiplication defined as follows. Given a \u00a3 A^ r b ( [G\\X ' ] ) for i \u00a3 {0,1}, define 7focro -Tf\\cfi \u2022 (ev), cr0 *' a\\ := ^2 1G(L \u00b0 ^0\u00b0)* (3\u20acA+(X)G . ic3'\\x,p) where 7fj : [ G ( G ) \\ X ( G ) ] 2 x X^ -> [ G ( G ) \\ X ( G ) ] is the projection on the ith factor and 7foo is the projection on X^G\\ Note that ao *' o\\ is G-invariant, hence it lies in the isomorphic image of A*orb([G\\X}) in A*(X-( G)) under <$>*xG. Let ao * a\\ be the preimage of ao *' a\\ under $*x G . Chapter 8. Group quantum cohomology 41 It was stated in [AGV02] that.qA*([G\\X]) is an associative algebra. 8.8 Invariant part of the group quantum cohomology By tensoring \u00ae*X,G with Q* (X, G) we get a Q-graded morphism of Q-modules qA*([G\\X])-+qA*(X,G). which will also be denoted by &*XG-Theorem 8.19. The morphism $*XG ^s a homomorphism of Q-graded algebras identifying qA*{[G\\X}) with the G-invariant part of qA*(X, G). Proof. For simplicity write lC{0) (resp. 77(0)) for lcl'G(X, 0) (resp. A \/ f (X , \/3 ) ) . Let q the quotient (xM){o,i,\u00b0o} [G{G)\\X^]i\u00b0^ x X^G\\ Then (i o Tfoo)* (TTS^O \u2022 Tfi^i \u2022 e v f \\]C(0)]mr^ = (<\u2022 \u00b0 Tfoo)* (TTO^ O \u2022 TFI^I \u2022 e v f pv\"(\/3)p r) = (\u00ab\u2022 o 7foo)*g* (g*7foCTo \u2022 9*7fiCTi \u2022 e v f pv\"(\/3)]w r) - ( to 7r G i 0 0)* ( T r ^ o ^ a o \u2022 ^ GAG^I \u2022 e v f pv-(\/?)]\u2122 r) which is by definition the coefficient of qG in (Q*X QO~O) *G {&X G a i ) - ^ Chapter 9 Associativity 42 In this section we address the problem of proving the associativity of the group quantum cohomology. As in the classical quantum cohomology, this is related to numerical equiv-alences of certain divisors on a moduli space, which in our case is A t 0 4 . We don't know how to prove it for a general G , but we can get partial results for cyclic groups and full associativity for Z 2 . In the classical quantum cohomology, one considers the space .Mo,4, which is P 1 , and hence any two points are rationally equivalent, providing the famous W D W equations. In our case, it is not true (as suggested in [JKK03]) that these ratio-nal equivalences in A - T 0 4 pullback from It is still possible that these divisors are algebraically equivalent, which also has been claimed in [JKK03], but we found the proof unsatisfactory, and hence we will state it as a conjecture. 9.1 Reducing the problem Suppose that a\\, a 2 and 0-3 are classes of A*(X,G) supported on {mi} x Xmi, {m 2 } x X\u21222 and {m 3 } x Xm3 respectively, where m i , m 2 and 1113 are elements of G. Then the associativity of the product in qA* (X, G) can be expressed as the relation (<7i * cr2) * 03 = (<7i * 0-3) * m 3 \u2022 <72. (9.1) Indeed [a\\ * cr2) * 0-3 = (cr2 * m 2 1 \u2022 <7i) * 03 by G-graded commutativity. By the above relation we have ( ) { 0 a , l a , * , l 6 , O O ( , } {0 a , l a ,OO a , 0b , l ( , ,CXD( , } T o o 0 X 7 r 0 t Let V a x 6 : {0a,la,ooa,0(,,li,,oO(,} {0 a , l a , * , l i , ,OO( , } be the natural evaluation maps, and let n : {o\u201e,i\u201e,Mfc,oot} ^ ^ ( G ) b e ^ projec-tions. Then the triple product (01 * 0 2 ) * 0 3 is equal to 0a, 0b where (3a and f3b run over A + ( X ) G . In particular the coefficient of qG is (^ooj* 7 r o a C T i 7 I \"L 0 '27ri i ,o-3(ev a | b )*A ! A ^ x b ( X , \/ ? ) where J7\u00b0xb{X,0) is the disjoint union of J7Gxb(X, pa, Pb) over all \/?<, and \/3b e A j \" ( X ) G such that Pa + Pb = P- Consider the forgetful morphism : ( x ( G ) ) { 0 a ' l o ' * ' 1 ' \" O O ! , } -> (A.(G )){Oa,i.,U,oo t} ) k t . ^ ( G ) j { 0 a , i o , i 6 l o o t } ^ X ( G ) b e t h e p r o j e c t i o n m o r p h i s m s . Then the above expression is equal to (*oo6)* ^ l a ^ l a ^ H 0 ^ 0 e v a | b ) * A ! 7?aXb(X, p) (9.2) To show associativity (see Equation 9.1) one has to show that the last expression is equal to (^ooi,)* flaG^laa^lbm3 1 ' a2(<\/> \u00b0 eV f l |b)*A ! (9.3) Consider the braid group B4 on four strands generated by Co\u201e,ia, Cia,ii> a n < ^ Cit,,oo(,- It has a unique action on (X( G ) ){\u00b0 a ' l a ' l i \" 0 O | >} such that 'TiCij \" x = ^i{x) \u2022 Kjx = KjHi{x) -j x ' x = a%x for k $ {i,j}- So for example: Cu,H \u2022 ( ( m O a . ^ O a ) , ^ , ^ ) , ^ , ^ ) , ^ , , ^ ) ) = = ( K , % ) , K m i i m i ' a V i 0 -a;ifc) . ( m i ^ z i j . t m ^ z o o j ) Chapter 9. Associativity 44 L e m m a 9.1. On intersection theory we have: Proof. We compute: l a , l 6 (^Oa^U^l^) = 7 r0 a C TlCl a , l i )7ri aCT2Cl o , l bTl i )CT3 = ^ O a O ' l (TflaCla.lb)* CT2 ( ^ l t C l a , l J * \u00b03 = 7ToaCTl ( f r i^ i , , \u2022!(,)* 0-2 (7t\"l a )* C T 3 = ^ O a ^ l (Mla-lt)* 7fli)CT2'ri t o show the associativity it suffices to t r Q 1 vir show that the class (<\u00a3oeva|{,)*A\" \\Jvaxb(X,(3)\\ is Ci a,i 6-invariant too. In next sections we will reduce this to the problem of rational equivalences of some divi-sors on A40 4 . We proceed as follows. First we study the behavior of the virtual fundamen-tal class under the gluing morphism JT^^X, (3) \u2014> M^(X, ft) where A = {O a, l a , 1;,, oo;,}. Then we study the behavior of the virtual fundamental class under the cutting edges mor-phisms TT^^X, (3) \u2014> J7Gxb(X, (3). These two results together with a conjecture about the Q rational equivalence of some divisors on A40 4 yields the associativity. 9.2 Gluing Suppose that E\\ and E2 are two flat projective G-curves over T, such that the action on every fiber is admissible and balanced. Let E be an etale G-scheme over T and let i\\ : E \u2014* Ei and i2 : E \u2014\u2022 E2 be two G-equivariant embeddings of schemes over T such that the images of i\\ and i2 lie in the smooth locus of E over T. Let m i : E \u2014> G and 7712 : E \u2014> G be the monodromies of E\\ and E2 along the images of t\\ and 12- Suppose that m i = m ^ 1 . Proposition 9.2. There exist a flat projective G-curve E overT, such that the action on every fiber is admissible and balanced, and G-equivariant T-morphisms l\\ : E\\ \u2014> E I2 '\u2022 E2 \u2014> E Chapter 9. Associativity 45 such that the following diagram of G-schemes over T E E2 commutes, and having the following universal property: given two G-equivariant mor-phisms f\\ : E\\ \u2014> X and f2:E2^X such that f\\\u00b0i\\ = f2 o t2, then there exists a unique G-equivariant morphism f : E \u2014> X such that commutes. Moreover the construction of E commutes with base change. Proof. Using the gluing of curves (non-equivariant) and its universal property (see [BM96]), one can glue G-curves and prove the above universal property. \u2022 Let Ai and A2 be two non-empty finite sets. Let g\\ and g2 be two non-negative integers. For i e {1,2}, let fc* G Ai. Consider the fiber product J J i A i k i \\ k 2 A 2 G - G x G where A(m) = (m, m 1 ) . By Proposition 9.2, we have a morphism of stacks Aiki\\k2A2 where A = (A\\ U A2) \\ {ki, k2}. This is called the gluing morphism along sections k\\ and k2. Using the universal property of gluing, we also have a morphism MGAlk.\\k2A2{X,l3l,p2) ^MG+g2A(X,f3i + P2) where the left hand side is defined as the fiber product M%lkl{k2A2(X,pup2) jtfiiAi(X,Pi) x jrfg\\M{X,p2) Chapter 9. Associativity 46 9.3 Compatibility of the virtual fundamental class under the cutting edges morphism Definition 9.3. Define the cutting edges morphism as the inclusion 7^ALKL]K2A2(X,pup2)^MGLTAI(X,p1) x M^A2(X,p2) in the last cartesian diagram. The general construction of the obstruction theory for stacks of the above form is given in the following. Definition 9.4. Given an Ar t in stack 9JT with an admissible balanced G-curve \u20ac \u2014\u00bb Tl, let M(X) be the stable locus of Morg l(\u20ac,SDft x X). Let TT : \u00a3 -> M{X) be the pullback of \u00a3, and let \/ : \u00a3 \u2014\u2022 X be the universal G-map to X. As in Section 4.1, we define the relative obstruction theory of M(X) over SD? as the pair {E,4>E) where \u00a3 : = R 7 r G ( u ; T \u00ae \/ * f i x ) [ l ] and 4>E '\u2022 E \u2014> -k^jq\/gjt * s * n e natural morphism (see Section 4.1). We've already constructed the obstruction theories and virtual fundamental classes for stack of the form M3IA(X, p) in Section 4.1. Now we can construct them for A4A1XA2(X,Pi,(h) and A4Alk1\\k2A2(X,Pi,P2). Using the above definition we just need to interpret MA1*A2{X, Pi,p2) and MALKL\\k2A2(X,Pi,P2) as stable loci in a stack of G -equivariant morphisms. The following is straightforward. Lemma 9.5. Let %RA1XA2 be the product o\/SJt^ and WIA2 \u2022 The universal G-curve \u00a3AXXA2 over WIAX XA2 is just the disjoint union of the pullbacks of the universal G-curves over TIA1 andWlA2- The stack 9JlAIXA2(X) := MOTGJ1a ^A (\u00a3A1XA2,X) decomposes into a disjoint union ^%XA2{X) = II mGMxA2{x,Pi,p2) Pi, & where the degree of the morphism is given by Pi and p2 on each component. Then MAIXA2(X,pi,p2) is isomorphic to the stable locus ofTlAIXA2(X,Pi,p2). Let 9JIyi1fci|fc2A2 ^e the product of 9RAI and WIA2- The universal G-curve \u00a3A1ki\\k2A2 over 971,4^ 11*2^ 2 * s obtained by gluing the pullbacks of the universal G-curves over SUl\/^ and WIA2 along ki and k2. The stack Tl^kl^^(X) := M o r ^ ^ {\u00a3A^\\k^A^x) decomposes into a disjoint union OTW^2P0 = II anWA 3(*.A,\/%) A , ft where the degree of the morphism is given by Pi and p2 on each component. Then MAlkl\\k2A2(X,Pi,pi) is isomorphic to the stable locus ofVJlG^KL^K2A^(X,Pi,p2). Chapter 9. Associativity 47 Using Definition 9.4, we have the virtual fundamental classes [MAlxA2(X,p1,p2)]vir 1 vir \u2022MA1ki\\k2A2(X^Pl^2) The following proposition says that the virtual fundamental classes are compatible under cutting edges morphism. Proposition 9.6. We have A! \\MAlxA2(X,p1,p2)]mr = p ^ ^ p ^ . A O J . Proof. The proof is a straightforward generalization of the proof of the axiom III in [Beh97a], where we replace locally free sheaves by locally free G-sheaves, pushforward of sheaves by equivariant pushforward, sections by twisted sections, the target space X by the inertia variety X^, etc. \u2022 Corollary 9.7. Let A! \\Maxb{x,p1,p2)}vir = \\Na\\b{x,pup2))1 and let 77\u00b0^ (X, P) be the closed and open substack of ~\\f\u00b0\\b(X, P) where the monodromies at the sections ooa, c i a , o e v 0 | 6 ) , [jr3(bm(X,p)]mr = (<\/>oeva|b), ^b^m(X,P) holds, then we have (a 1 * (72) * cr3 = CTI * [p2 * cr 3). Proof. As the group quantum cohomology is a G-graded G-algebra, note that (o\\ * u 2 ) * CT3 is supported on the twisted sector corresponding to m ^ 1 = 01^2^13. It follows that in the equation Equation 9.2 (resp. Equation 9.3) we can restrict the class (<\/> o ev a | 6)* A! p7\u00b0xb{X, \/?)] V W to Xmi x X 1 \" 2 x Xm* x Xm* (resp. to X m i x Xm3 x I m 3 _ 1 r Q m -j vir X\u21224) which is equal to (<\u00a3\u00b0ev a|6)* A a | b {X, P) (resp. to (4>oeva]b by the above corollary. The result follows from Section 9.1. \u2022 ' X vir Chapter 9. Associativity 48 9.4 Compatibility of the virtual fundamental class under the gluing morphism We consider the morphisms <\/\u00bb: MA(X,p) ^WlA ip' : MAlk^k2A2{X, Pi,P2) \u2014> 9^Ai*i|FC2>i2 that forgets the G-equivariant map into X. We also have the stabilization morphisms s:WlA^> MA s' : W l A l k i \\ K 2 A 2 ~~* \u2022 ^ A 1 k i \\ K 2 A 2 that stabilizes the G-curve. We have the gluing morphisms gl(X) : MAlH]k2A2(X, Pi,fo) -> MA(X, Pi + p2) gl : MAlkl][k2A2 -> MA il:WlAikl{k2A2^WlA. Lemma 9.9. The diagram MAlkllk2A2(X,pf(-^MA(X,p) Aiki\\K2A2 gl WIA is cartesian, where horizontal (resp. vertical) arrows are gluing (resp. forgetful) mor-phisms, and where MAlki{k2A2(X,p):= MAik l\\K2A2 \/3i+\/32=\/3 MAlkl\\k2A2(X, p) Moreover iX \\MA(X,p)]vir Proof. We first prove that the diagram \\K2A2 (x, p) \u2014 m A ( x , p) L 971 Axki\\K2A2 gl is cartesian, where the horizontal arrows are the gluing morphisms and the vertical arrows are forgetful morphisms. Let T be a scheme and let f: T \u2014> WlA(X, p) and (ei , i2) '\u2022 T \u2014> ^\u2022A1ki\\K2A2 D e m o r p h i s m and let 6 be the natural transformation making the diagram T e-+WlA(X,p) L Alkl\\K2A2 Chapter 9. Associativity 49 commute. We need to show that there exists a unique morphism T \u2014> ^IR Axk^k2 A2[X, P) commuting with the gluing morphism and the forgetful morphism. By definition of 3JI j4 ifc1|fc2^2(X\", \/?), we need morphisms f* : T \u2014\u2022 SDT^pQ for z G {1,2} such that evj^ ofx = i o ev^2 of2 where i is the involution on X^G\\ and such that locally the degrees of \/ n and \/ f 2 add up to p. Let := Eti, \u00a3 C i := E C i and cr f i i f c := cr e i i f e. Define \/ f \/ by where i$ are the inclusion into the glued G-curves. This is the desired morphism and it is unique with the above properties. The first statement follows from the fact that the gluing morphism is representable and hence preserves stability. The second statement follows from the first and [BF97, Proposition 7.2]. \u2022 Consider the diagram MA1ki\\k2A2(X>P) ' \u202221-03 \u2022MAlk1\\k2Aa gl \u2022MA(X,p) \u2014 i -9JU \u2022MA where every square is cartesian (by Lemma 9.9), and where h and I are uniquely defined by universal property of fiber products and the commutative diagrams MA!kl\\k2A2(X>P) L Aiki \\ k 2 A 2 ^ A i k i \\ k 2 A 2 MA(X,p) WIA \u2022MA Proposition 9.10. We have M A^ki\\k2A2\\ (x,p) =gmMA{x,p)]mr). Proof. As both MAlk^\\k2A2 and MA are smooth of pure dimension, the gluing morphism is a local complete intersection in the sense of [Ful98] and [Vis89] and has a well defined orientation [gl] in bivariant intersection theory (see [Ful98, 17.4] and [Vis89, Section 5]). Chapter 9. Associativity 50 We first claim that one has the following relation in the bivariant intersection theory f, [gl] = s* [gl]. We adapt the proof of [Beh97a, Proposition 8]. This claim follows from the three facts: I is birational, s is flat and gl is a local complete intersection. We first show that I is birational. That the restriction of I to the open dense substack A4Alkl jfc2>i2 of 9RAiki\\K2A2 is an isomorphism onto its image is clear as the stabilization morphism restricted to A 4 A l h 1 ^ 2 A 2 is the identity. We only need to show that I is surjective. A n object of A^Aikt-\\K2A2 o v e r SpecC is a pair ( e i , e 2 ) where for i G {1,2}, e , is an object of M.Ai such that the monodromy of Etx at oti is inverse of the monodromy of Et2 at : gl{t\\,t2) \u2014 s(f). Hence we have a diagram Etl II Et2 ~ *\" Et where e is some object of MA, and where s contracts the unstable components and t identifies the orbits G \u2022 Eti completing the above diagram. It is clear that the Sj's are contracting the components to be contracted. This produces an element ( f i , f 2 ) of T \\ A l k 1 ] k 2 A 2 s u c n that i ( f i , f 2 ) is isomorphic to the object ( ( e i , e 2 ) , 4 > , f ) of 03. The claim follows from the birationality of I. The rest of the proof is as in [Beh97a, Axiom V] : gl ! {MA(X,P)]vir = (s* [gl]) {MA(X,P)]vir = (l,[i\\\\)\\MA(X,P)]vir = KiX\\MA{X,(3)]vir h* MA lk l \\ K 2 A 2 (X,P) where s* [gl] = U |gl] was proved above. The following is a straightforward consequence of the above proposition. \u2022 Chapter 9. Associativity 51 Corollary 9.11. We have K \u00a3 [A 7 ^ 1 ^|^y l 2 (^\/3 i ! \/32) ]\" r =A7 ( 5 0 ! ( [>l A (X ) \/ J ) ]^) 01+01=0 where 77(gl) is the composition j7Alk^k2A2 \u2014> MAlk1]k2A2 \u2014> MA, and h! : 21' \u2014> . \/ V ^ X , \/ 3 ) is \u00a3\/ie pullback ofJT(gl). 9.5 Associativity in group quantum cohomology Note that A7^ jj, maps isomorphically onto the subset of G4 consisting of quadruples (mi, m2, m3, m4) such that mim2iii3m4 = id by the map ((mo^mi^mooj , (m0b, m i 6, m^)) ^ (m0a, mi a, m^, m^). The braid group B4 has a natural action action on G 4 defined as follows: Co0,i0 \u2022 (m0o,mio,mi i ),mOOi)) = Kmi,\\ 1,mo.,mi l,moo l) Cia.it, \u2022 (moaImia>mi!,>moo6) = (mo^m^m^m^Smi^mooJ Cib.ooi, \u2022 (mo^mi^mi^mooj = ( m 0 ( l , m i a I m i l m O O ! ) m ^ 1 , m l b ) and this action preserves A \/ ^ b under the above inclusion. It also has a natural action on defined as follows. Given an object e = (E,ooa,aia,aib,o00b) of A ^ ' m , where m = (m 0 o,mi o,mi i ),m o oJ G G 4 , let Coa,ia \u2022 e = (E,m0a \u2022 (Tia,croa,(rib,eTool>) Cia.ib' c = (E,o-Oa,mia \u2022 (7ib,aia,aoob) &b,oob \u2022 c = (E,a0a,aia,mib \u2022 e2 of is given by a G-morphism a of the underlying curves, then the morphism \u00a3 \u2022 f is given by the same G-morphism a, for any ( e B4. There are obvious natural isomorphisms Ci(C2\u00ab) \u2014 (GC2)e f \u00b0 r 0 \u00a3 - B 4 ensuring we have an action of a group on a stack. Even though gl(m) G MA'm, the gluing morphism gl : A 7 ^ 6 \u2014> M\u00b0A is not B4-equivariant under these actions. This can be seen in the simplest case where the group G is trivial. The right hand side is isomorphic to MoA. The braid group B4 acts on .\/Vfo,4 by permuting the markings, and has no fixed points. The left hand side A7^j6 is just a point, hence the morphism gl cannot be equivariant. On the other hand, it is equivariant up to rational equivalence as MOA \u2014 P 1 - We can speculate that gl is equiv-ariant up to rational equivalence for a general G . Unfortunately M A m is far from being P 1 in general. In fact, in general it is not even connected and the connected components aren't necessarily rational curves. To see this, take m = (mi, m2,1113, id), then by a The-orem of Abramovich-Vistoli, the forgetful morphism M\u00b0'm \u2014> j^\\G'mi'm2'm2 j s a u n i V e r s a l G-curve, and the fibers are curves whose connected components can be of arbitrary genus Chapter 9. Associativity 52 by the Hurwitz formula. In later sections we will show some examples where gl is equiv-ariant up to rational equivalence. For now, we will state the conjecture and show how the associativity follows from it. As before let A \/ ^ m be the locus where the sections have monodromies given by m, which in this case is just the point ( e (mi,m2,mf2 )' e(mi2,m 3,m 4) ) where m 1 2 = m i m 2 , and where e ^ m i m 2 m - i - j and e ( m i 2 m 3 m 4 ) are points of A \/ Q as defined in Section 7.1. Let e : Af\u00b0\\b - \u00bb J ^ A m =>1 (e(m1,rti2,m1\"21)' e(mi 2,m 3,m4; Conjec ture 9.12. The class g l , [A7^jb] is B^-invariant in A i ( A 4 G ) . More \u2022precisely, given m G G4 such that m i m 2 m 3 m 4 = id, we have C*e* [m] = e* [C 1m] for all C G B4. We will prove this in certain particular cases in the next three sections. For now we will state the important consequence of this conjecture, namely the associativity. The braid group B4 has a natural action on A4A{X, j3), defined in a similar way to the action on M^, such that A4\u00b0_(X, p) \u2014> 7\/1^ is equivariant. The morphism M^iX,0) \u2014\u2022 (x( G ))^ 0 o ' l o ' 1 '\" o \u00b0 ! ^ is equivariant as well, where the action on (x-(G)^f0a'lo>1'>>00&} j s g j v e n Q -| vir A4A{X,P)\\ is invariant under the action of B4 as it is invariant under permutation and translation of the markings. In fact for any ( \u20ac \u00a3 4 , c* JtA^{x,p) M M^\/~lm(X,P)^MG\/\"(X,p). ~* I 1m 1 1 as the action of \u00a3 restricts to an isomorphism P r o p o s i t i o n 9.13. The above conjecture implies that ( o ev Q | b = ev^ og l (X) . Since evA is B4-equivariant, we need to show that c f g l p O , fttff{X,pj gl(X)JA7%bm(X,P) ~^\\b~1 m(X,P) g l X [ A ^ b m ( X , \/ 3 ) ] g i : g l ! [ A ^ ' ' m ( X , \/ 3 ) ' ^ S * g l , [ A 7 G | b m ] n [ A 4 G ' m ( X , \/ ? ) We have by Corollary 9.11 Chapter 9. Associativity 53 where gl' is the pullback of gl under sotp, and the last step is by [Ful98] [Example 6.3.4]. by Conjecture 9.12 and We have C*gl* [ A ^ 6 m ] = C*C [m] = e. [C'm] = g l , [iJ^f m C* \\A4A' (X,0)\\ = MA (X,(3) by the above remark. Moreover ip and s are jE?4-equivariant, which proves the result. \u2022 By the last proposition and the Section 9.1 we have: T h e o r e m 9.14. The Conjecture 9.12 implies the associativity of the group quantum co-homology. In particular if m \u20ac GA is such that mim2m3ni4 = id and if o\\, a2 and a3 are classes of qA*(X,G) supported on the twisted sectors corresponding to mi, m2 and m 3 respectively, then the equality Cia,ibe* [m] = c * [Ci^i,,111] in Conjecture 9.12 implies (CTI * a2) * a3 = ai * (1J2 * 0 3 ) . 9.6 Smooth locus of Tt^A ->\u2022 WlgA We begin by studying the ramification locus of the morphism WlGA \u2014> 37ts>J4 that sends a G-curve to its quotient. L e m m a 9.15. Let E \u2014\u2022 SpecC be a nodal curve with admissible generically free G-action, and let ITE : E \u2014* C := E be the quotient. Let flE\/C : = = ^ f i \/ ^ ^ C and k = | Stabc;(p)|. Then for any p e sp(E), we have WE\/C,p) = * - 1. if p is a smooth point and ifp is a node. In particular TTE '\u2022 E \u2014> G is etale except at points with non-trivial stabilizers. Proof. The result is trivial for smooth p e sp(.E). Suppose p is a nodal point, and q = ITE{P) which is also nodal as the action is admissible ( [Miz05]). Let E (resp. G) be the local scheme S p e c 0 \u00a3 ] P (resp. SpecOc ) g ) , where C B , p \u2014 SpecC[[a:,y]]\/(xy) is the completion of the local ring \u00a3>E ) P, where x and y are both eigenvectors for the action of StabG(p), and similarly for Oc,q \u2014 SpecC[[u, v]]\/(uv), where B H I ' and v 1\u2014> yk under the quotient SpecC^p \u2014* SpecOc,?-Then fig ~ C[[a:]] \u00a9 C[[y]], and the derivation OE \u2014> fig induces the C-derivation dE:C[[x,y]]\/(xy)^C[[x]]\u00aeC[[y]] x '\u2014> dx :\u2014 (1,0) 2\/ <-> dy := (0,1). Similarly for fig ~ C[[u] \u00aeC[[v]}, and dc- Hence the morphism fig \u2014> fig of Og-modules, induces C[[u]] \u00a9 C[[v]] \u2014> C[[x]] \u00a9 C[[y]], the unique morphism of G[[w,t;]]\/(m;)-modules such that du i-> kxk~1dx and dv H - \u00bb kyk~1dy. It follows that f i n \/ c -.= nE\/w*End ~ c i M ] \/ ^ - 1 ) \u00a9 c i t y ] ] \/ ^ - 1 ) . \u2022 Chapter 9. Associativity 54 Theorem 9.16. Let M G A be the locus in M ^ A \u00b0f admissible G-curves whose nodes (if any) have trivial stabilizers in G. Then TT': M . G A \u2014> M G , A is etale. Proof. Let M -.= M G A , M\u00b0 := M G A and M := M G , A - Let t : T -> T' be a square 0 extension by a quasi-coherent ideal J. Let c be a family of M over T and c be a family of M. over T'. Suppose there is a morphism (a, t) : e \u2014> c over t. Consider the diagram y \u2022r where \u00a3 j is the unique deformation of S c to T' (the existence and uniqueness follows from etaleness of E c \u2014\u2022 T) . We will study the obstruction to completing the above diagram with the doted lines, such that f : = (E^,pf, \u00a3 f , C f ) is an object of M \u00b0 , and such that \u00a3 * f = e. For simplicity let E := Ee and E := Et. The obstructions to extending the above diagram lie in the group ExtG'2(\u00a3lE\/E,I) where 1 := v\\ J \u00aeoBt Z(E , ) and where I ( S e ) is the ideal sheaf of E e in Ee. Using the Grothendieck spectral sequence applied to the functor H o m G = H G ' \u00b0 oliom, to show that the above group is 0, it suffices to show that BG\/\u00a3xi\\SlE\/E,l)=Q uG'1\u00a3xt\\nE\/E,i) = o HG>2Hom(flE\/E,I)=0. Using Grothendieck flat base change theorem, it suffices to assume that T = SpecC. The group HE'2 Hom(flE\/E,I) vanishes as E is of dimension 1, and RE'1 \u00a3xt1(n\u00a3,^,X) van-ishes as flE\/E is concentrated at the nodes and markings. Next we show that \u00a3xt2(flE^E, J) 0. The exact sequence of G-O^-modules 0 -> 7 T * % ^ QE - 4 fl E\/E where TT : E \u2014> E is the quotient, induces a long exact sequence \u00a3xt\\TT*Q.Ea) *\u00a3xt2(nE\/E,l) Chapter 9. Associativity 55 where \u00a3x t 2 ( f2# ,Z) = O a s \u00a3 - > SpecC is a reduced complete intersection, that is flE has a two step locally free resolution. If p G sp(E) is a node with trivial stabilizer, it follows by Lemma 9.15 that \u00a3lE\/Ep = 0. If p G sp(E) is a smooth point, q = 7r(p) is a smooth point and irEQE is a free O^-module in a neighborhood of p, and \u00a3xt1(Tr*ClE,X)p = 0. The two above cases imply \u00a3xt2(flE\/E,l) = 0 . \u2022 9.7 Zfc-covers of P 1 by P 1 Let G = Zfc =< r >. Suppose v\\ = T, v2 = r _ 1 and v3 \u2014 id. Then M%>k$ \/ \u2014 SpecC. Proof. The dimension of is 0. Moreover the objects of M^k3\" over SpecC are isomorphic to the fc-fold cover of P 1 with 3-markings where two are totally ramified. This cover has no non-trivial automorphisms, and the result follows. \u2022 Suppose v\\ = r , v2 = r _ 1 and v$ = i>4 = id. We have an isomorphism MQk4 rz. P 1 such that the projection 7r : MQ^\" \u2014> Mo,4 \u2014 P 1 induces a degree k morphism P 1 \u2014\u2022 P 1 with two totally ramified points. Proof. We know that M^l\" is a smooth Deligne-Mumford stack of dimension 1. Let e := (E,J2,a) be an object of . M o 4 \" over C, and let e := (E,o) be the projection onto A4o,4- Since E = E\/Zk is a genus 0 curve, and the quotient E \u2014> E has two ramification points o 1 and o2 of ramification degree k, by the Hurwitz formula for nodal curves we get that E has arithmetic genus 0. The automorphism group of c is trivial as there is no automorphism of a genus 0 nodal curve with 4-marked points. It follows that has no stacky points and therefore is a smooth curve. Since each smooth stable curve of Mo,4 has k non isomorphic Z^-covers with mon-odromies i\/i,...,i>4, the morphism \u2014> MOA has degree k, and it is a degree k cover o f P 1 . _ The action of Z*, on by translation of the 4th section, makes A4Q 4 ' \" into a Z f e-cover of P 1 . Next we show that this morphism has 2 totally ramified points. Let T = ( P 1 , 1 , 0 , 0 0 ) be the 3-marked Z^-cover where Z& acts with fixed points 0 and 0 0 of monodromy r and T - 1 respectively. We glue two copies of T along 0 of the first and 0 0 of the second, and we can order the markings in two non-isomorphic ways. We get two point of A4Q 4 \" which are fixed under Zk-Finally we show that there can't be more ramification points. By Theorem 9.16 the only possible ramifications are at points c := (E, E , o) where E has a node with a non-trivial stabilizer. Let x G E be a node. Let E \u2014 E\\ U E2 be the decomposition of the quotient E into its two irreducible components, and let E\\ (resp. E2) be the closed subvariety of E above E\\ (resp. E2). Suppose <7j and Oj G E\\ for i \/ j, then the monodromy of x on E\\ must be ( ^ t ' j ) - 1 . In the case ^ or Uj is id, we get one of the two points of the last paragraph. Otherwise the monodromy of x is id and the curve E has no nodes with non-trivial stabilizer. It follows that MQ K\u00b1 ~ P 1 . \u2022 Chapter 9. Associativity- be T h e o r e m 9.17. Let act on a smooth projective variety X. Let o\\, o~2 and 03 be classes of qA*(X, Zfc) supported on the twisted sectors corresponding to id, id and r respectively. Then (a\\ * a2) * o3 = o\\ * [a2 * 03). Proof. Let m = (id, id, T , T _ 1 ) . Then for any \u00a3 G B4, C, \u2022 e m and e^.m are both points in A4Q 4'^ m ~ P 1 , so they are rationally equivalent, and the result follows from Theorem 9.14. \u2022 9.8 Z2-covers of P 1 by genus 1 curves Let G = Z2 =< r >, v\\ = ... = t\/4 = T. The morphism 7r : M Q 2 ^ \u2014> Mo,4 is an etale Z2-gerbe. Proof. We know that M Q 2 ^ is a smooth Deligne-Mumford stack of dimension 1, and that 7r is etale over the locus M^2^ of curves with nodes whose stabilizer is trivial by Theorem 9.16. We will show that M^f = M^f \u2022 Let e := (E, E, Y is a G-morphism of G-schemes. There is a pullback functor for sheaves f~l : S h G ( Y ) \u2014> S h G ( X ) and a pullback functor for modules \/ * : q C o h G ( X ) \u2014> q C o h G ( y ) (see [Gro57]). If X is noetherian or \/ is quasi-compact and separated, then there is a pushforward functor \/* : q C o h G ( X ) -> q C o h G ( Y ) . If the G-action on Y is trivial, one also has an invariant pushforward functor fG : q C o h G ( X ) \u2014> qCoh(F) defined on objects by fGT{U) = ^ r ( \/ - 1 C \/ ) G for an open U CX. Assume that G is a finite group and X is a G-scheme such that the quotient Y = X\/G exists as a scheme. Let IT : X \u2014> Y be the quotient morphism. It is proper, and by the preceding paragraph we have an invariant pushforward TT g : q C o h G ( X ) - \u2022 qCoh(y). We will now list without proof some properties of 7r G . Lemma 1.1. The canonical injection ~KGT \u2014> TT^J7 and the canonical morphism 7r*7r* .F \u2014> T induce a morphism of G-Ox-modules 7r*7rG.F \u2014> -K*-K*T \u2014> T that is an isomorphism away from points with non trivial stabilizers, thus injective. Appendix A. Equivariant Riemann-Roch for Nodal Curves 58 Def in i t i on 1.2 ([BorOO], 2.11). A G-scheme X is called loca l ly reduct ive if for each point P G sp(X\"), the order of the stabilizer Gp is invertible in Ox,p-L e m m a 1.3 ([BorOO], 2.12 and 2.29). Suppose that X is a locally reductive G-scheme such that the quotient TT : X \u2014> X\/G = Y is flat. If T is a locally free G-Ox-sheaf of finite rank, then i r G T is a locally free Oy-module of the same rank and TZ1-KG{!F) = 0 for i > 0. A . 3 Cohomology Let Yx{Ox) (resp. Tx{Ox)) be the ring of regular functions (resp. G-invariant regular functions) on X. Note that Tx{Ox) is naturally a G-ring. We consider the following two functors Yx : qCohG{Ox) - qCohG{Tx{Ox)), TGX : q C o h G ( 0 x ) -> qCoh{TGx{Ox)) where the first one is the functor of global sections and second is the functor of G-invariant global sections. The two functors are left exact and the category qCoh G (C?x) has enough injectives ([Gro57]), hence we have the right derived functors ffx := P J I * : q C o h G ( O x ) -> qCohG {V x{0 x)), H G , i := R ' r g : q C o h G ( C x ) - _ q C o h ( r \u00a3 ( 0 j r ) ) . L e m m a 1.4. Suppose that X is a locally reductive G-scheme that admits a quotient Y = X\/G. Then H* y(7rf JO ~ T3%\\F) for any locally free G-Ox-module T on finite rank and for all i > 0. In fact there is a natural isomorphism between B.x 0 7 r G | p G ( o x ) and H G ' 1 |pG(e>x). Proof. We can write rGx = r Y O i t G and the result follows from the Grothendieck spectral sequence ([Gro57] 5.2.3) and from Lemma 1.3. \u2022 L e m m a 1.5. Suppose that X is a G-scheme such that \\G\\ is invertible in Tx(Ox)- Then In fact there is a natural transformation between T G o and H 0 \/ . Proof. We can write r G = r G o Tx where TG : qCohG(Tx(Ox)) -> qCoh(rG :((!?x)) (the functor taking G-invariants) is exact, and Rl T G = 0 for i > 0. The result follows from the Grothendieck spectral sequence. \u2022 Appendix A. Equivariant Riemann-Roch for Nodal Curves 59 A.4 Representions of finite groups and their characters Let G be a finite group and A; be a field whose characteristic does not divide the order of G. Denote by Rep f c G the category of ^-representations of G and G-equivariant fc-linear maps, and by Irr^ G the set of irreducible representations of G . Given V a representation of G , denote by Xy '\u2022 G \u2014> k the corresponding character. The characters of G generate the character ring Chfc G . We have a pairing on Chfc G given by for any X\\ and X2 G Chfc G . If V and W are representations of G and V is irreducible, then (XW,XV)Q is the multiplicity of V in W. The inverse i : G \u2014> G induces by composition an involution of Chfc G , denoted by X ^ X Y . Note that ( A i ^ , A r 2 ) = (X1,XVX2). Let i f be a subgroup of a finite group G . Given a representation V of G we denote by Res G V the natural representation of H on V given by restriction. Given a representation W of H, the space G x W is a right fl-representaion with action (g, v) \u2022 h = (gh, h~l \u2022 v) and the quotient is a representation of G denoted by Indff W. The operations Res^ and Ind^ are functors between the categories Rep f e G and Rep f c H. They induce group ho mo morphisms between the underlying groups of character rings of G and H, that will be denoted by the same symbols. They are adjoint in the following sense: (Xu Indg X2)G = (Resg X\\, X2)H (A.1) for X\\ e Chfc G and X2 G Chfc H. We will list some properties of characters. \u2022 For any representation V of G Xv(id) = d i m K (A.2) \u2022 If k[G] denotes the standard representaion of G , then (A.3) Velrrk G \u2022 For any X G Ch f c G (A.4) V e l r r f c G L e m m a 1.6. If H is an abelian subgroup of G and X \u00a3 Chfc i f , then \\ndGH{XvX) = X{id)Xm. V e l r r f c H Appendix A. Equivariant Riemann-Roch for Nodal Curves 60 Proof. The irreducible characters of G form an orthogonal basis for Chfc G under the pairing (\u2014, \u2014}G, hence by Equation A . l we have Y lndG(XvX) = Y \u00a3 (IndGr(XvX),Xw)GXw VElTikH W\u20aclTTkGVeliTkH = ^ Y, (XVX, R e s G XW)HXw Welrvk G Velvik H = \u00a3 Yl (Xv,Xv Res\" XW)HXW \u2022 W e l r r f c G V e l r r f e H The irreducible representations of the abelian group H are one dimensional, hence by Equation A.2 to Equation A.4 IndGr(XvX)= (Xy Res% Xw) (id)Xw VeliTk H W&xtk G = A' v ( id ) dimWXw Welvrk G = X{id)Xk[G]. \u2022 A. 5 K- theory Let k be a field whose characteristic does not divide the order of G. Let X be a finite dimensional G-scheme, projective over k and such that the structure morphism X \u2014> Specfc is G-invariant. It follows that I is a locally reductive scheme and that | G | is invertible in Tx(Ox) = Tx(Ox) = k. Given an abelian category jrf, let K{srf) be the Grothendieck group of srf. The tensor product of G-Ox-modules induces a ring structure on K (PG(X)). Let T be a locally free G-O^-module of finite rank. B y a theorem of Serre ([Gro61b], 2.2.1) WX(T) are finite dimensional vector spaces over k. Moreover they vanish for i > d i m X by a theorem of Grothendieck ([Gro57], 3.6.5). The same is true for HG '*(J\") ~ B. x (T)G. It follows that the two functors Yx : PG(X) - PG(A;) TG : PG(X) - P(fc) induce ring homomorphisms on K-theory K(TX) : K ( P G ( X ) ) - K (PG(fc)) ~ Ch f c G K ( r G ) : K ( P G ( X ) ) ^ K ( P ( f e ) ) ^ Z Appendix A. Equivariant Riemann-Roch for Nodal Curves 61 We define Xx (resp. X^) to be the composition of K(rx) (resp. K(TX)) with the isomorphism X : K (PG(fc)) ~ Chk(G) (resp. dim : K (P(k)) ~ Z). Given an object F of PG{X) we will write XX,T (resp. XGT) for XX([F]) (resp. #G([JF])). In fact we can express Xx in terms of XG. Lemma 1.7. Let F be an object ofPG(X). Then V 6 I r r f c G where F\u00aeVy is by definition F\u00aeox ^V where V is the free G-Ox -module corresponding to the constant G-vector bundle X x V. Proof. Xx,r = ^2 (XX,F,XV)XV v e i n * G v e i n * G = \u00a3 (Xx,r\u00aevv,id)Xv-v e i n * G We used H ^ ( J r ) \u00ae V v = H^(JT \u00ae V v ) where V is a representation of G. B y Lemma 1.5, we have that i>0 = \u00a3 ( - l ) M i m f f x ( \/ - \u00ae V v ) G i>0 = \u00a3 ( - l ) i d i m H G ' i ( J : - ( 8 ) y v ) i > 0 \\>G \u2014 A-X,T\u00aeVW i and the result follows. \u2022 A.6 The smooth case Let G be a finite group and k be an algebraically closed field whose characteristic does not divide the order of G. Let E be a smooth projective G-curve over k. In this section, we will compute the homomorphism XE just as in [EL80] and in [Bor03]. Let P G sp(.E) and suppose F is a G-C^-module such that Fp is a free Gp-Op,p-module of finite rank. Define Xpp := Xpp G Oh*, Gp XGr : = d i m F G p G Z where Fp = k \u00aeoE P Fp, which is naturally a fc-representation of Gp. Appendix A. Equivariant Riemann-Roch for Nodal Curves 62 De f in i t i on 1.8. If the set of points with trivial stabilizers is dense in E we call the G-action generical ly free. Suppose E is endowed with a generically free G-action. If P e E is a smooth point, then the stabilizer G p is cyclic as it has a faithful one dimensional representation on the tangent space of E at P. L e m m a 1.9. Let E be a smooth projective curve over k with a generically free G-action and let TT : E \u2014* C be the quotient. Let F be a locally free G-0E-module of finite rank. Denote by QE the sheaf of Kahler differentials of E. For each P e E let ep = \\Gp\\. Then 1 1 e p _ 1 degTrfjT = \u2014 deg F - \u2014 V Y~ j- XGp n i 1 1 1 1 P\u20acE j=0 Proof. The canonical morphism TT*TTGF \u2014> F is injective and we have an exact sequence of G-Ofi-modules 0 -+ -K*-KGF ->F^K^0. (A.5) where 1Z is a G-C^-rnodule supported at the ramification points of TT : E \u2014> C. Thus deg ir?F = \u2014!\u2014 deg w*wGF deg7r = j ^ | (deg .F -deg7e ) = - L ( d e g . F - \u00a3 l ( f t p ) ) ' ' PEE where l(TZP) is the length of the CV^-module KP. Let P \u20ac E and Q = TT(P). B y localizing and completing Equation A.5 we have 0 -+ dE,P \u00ae 6 c q F%p ^ F P ^ K P ^ 0. (A.6) Now O E , P \u2014 M M ] where u is an eigenvector for Gp, spaning a faithful representation with character XP:QE, which implies OC,Q \u2014 O g p \u2014 fc[[uep]]. We have Fp c\u00b1 OE,P <8>fc Fp ~ k[[u]] ~~k Fp- The inclusion in Equation A.6 is just the homomorphism k[[u]} \u00aek[[ueP\\] {k[[u}] \u00ae f c FP)Gp -> k[[u}} \u00aefe FP coming from the inclusion (k[[u]\\~~fc FP)Gp -> k[[u}} \u00aek FP by tensoring by k[[u]}. The group Gp is cyclic, hence the irreducible representations of Gp are all one di-mensional and their characters are homomorphisms Gp \u2014> k*. We can talk about the eigenspace Vx of some character X and a representation V of Gp. The representation ^ E , P of Gp is faithful and thus powers of XptnE run over all irreducible characters of Gp. Appendix A. Equivariant Riemann-Roch for Nodal Curves 63 It follows that (k[[u]] \u00aek FP)Gp = 0 k[[u]]xv\u00aek(FP)Xv Velrrk GP ep-l = 0 kM\\Xi \u00aek {Fp)xi 3=0 ep-l = 0 k[[ue\"]] \u2022 V> \u00aek {FP)x3 and e p - l <5 \u00a3,P \u00ae A C Q ^ = 0 k[[u}} \u00ae M [ U . P ] ] \/ c [ [ U e \" ]K \u00ae f c ( F P ) . e p - l < V? > \u00aek{FP). \"P,Q 0 u J \u00aek{FP) j where < wJ > is the ideal in k[[u]] generated by v?. It follows that. ep\u20141 Hi d i m ( F p ) ^ 3=0 e p - l ( A : p , n v . ; t > , ^ ) G p j=o \u2022 Now we compute X9 and X E . Proposition 1.10. Let E be a smooth projective curve over k with a generically free G-action and let it : E \u2014\u2022 C be the quotient. Let F be a locally free G-0E-module of finite rank. For each P G E let ep = \\Gp\\. Then 1 1 1 1 P V v to get XBf = E \\W\\ d e g ( ^ \u00ae ^ ) + XcVc \u00ae ^ V ) - T^T E E J \" <4,fiv , ^ \u00ae V v ) G p ) Xv V e l r r f c G \\ 1 1 1 I P G B j = 0 ' E J = E U T7T d e ^ + xc,Oc r k f ) d k V 4 E b ' (4nv><*V R E S G P ^ V ) G P 1 * v M l \/ I I P C ;? \u201e\u2022_ n X) (Itesg i , AV ) 4 i n B A>^> G p AV= \u00a3 (Xv,lndGp^XPtQEXP^)GXv VehrkG V 6 l r r F C G \u2022 A.7 Normalization of a G-curve Let \u00a3 be a curve and n : E \u2014> E be its normalization. Any G-action on i? induces a unique G-action on E making n into a G-equivariant morphism. L e m m a 1.11. Suppose E is a projective G-curve and 7r : E \u2014> C is the quotient. Let rjB '\u2022 E \u2014> E be a normalization of E. Then the induced morphism E\/G \u2014> C is a normalization of C. Proof. First note that given a smooth dense open subset C ' C C then the normalization of C is the unique smooth projective curve C (up to unique isomorphism) containing C' as a dense open subset together with a morphism r]G : C \u2014> G whose restriction to C' is just the inclusion. Let n : E\/G \u2014* C be the induced morphism from the G-invariant morphism \u00a3\" \u2014> 15 \u2014> G . Let E\u00b0 be the smooth locus of E. Then E\u00b0\/G is a smooth and dense subset of C . But this inclusion factors through E\/G \u2014> G . And as . E \/ G is smooth and projective, it is the unique smooth projective curve containing E\u00b0\/G as a dense open subset, thus it is the normalization of G . \u2022 Appendix A. Equivariant Riemann-Roch for Nodal Curves 65 It follows that we have a commutative diagram E *E C ^ C where TJE and r\\c are normalizations and TT and TT are quotients. A.8 Admissible action If \/ : Y \u2014> X is a G-morphism of G-schemes, then it induces a canonical monomoprhims of stabilizers GQ \u2014> Gj^ for every Q G Y. Definition 1.12. A G-morphism \/ : Y \u2014\u2022 X is called fully faithful if for every point Q \u00a3 V the canonical monomorphism GQ \u2014> G J ( Q ) is an isomorphism, in which case XQJ'T = xf(Q),r a n d {XQ,fT,XQ,fc)GQ = ( Xf(Q),F, Xf(Q),c)Gf(Qy Definition 1.13. A G-action on a curve E is called admissible if the normalization rj : E \u2014> E is a fully faithful G-morphism. Lemma 1.14. Let G be a finite group and k be an algebraically closed field whose char-acteristic does not divide the order of G. Suppose E is a nodal curve with an admissible G-action defined over k. Then the quotient C = E\/G is a nodal curve, and the projection TT : E \u2014> G sends nodes to nodes. Moreover if E\u00b0 and C\u00b0 are the smooth loci of respectively E and C, then \\ c\\ c\u00b0\\ = w\\ \u00a3 |Gp|-1 1 P\u20acE\\E\u00b0 Proof. Let P G E be a node and let Q\\ and Q2 be the two points in E above P. Since 77 is a dominant morphism of algebraic varieties, we have an injective G-morphism OE \u2014> V*\u00aeE of G-O^-sheaves. Localizing at P and taking the completion we get an exact sequence 0 -> k[\\u, v]]\/{uv) t k[[u)) \u00a9 k[[v]] ^ k ^ O (A.7) where OE,P \u2014 k[[u,v]]\/(uv), 0EQ ~ k[[u]] and O E Q 2 \u2014 k[[v]], the homomorphism is given by p(u,v) (p(u,0),p(0,vj) and S by (p(u),q'(v)) H-> (p(0) - (7(0)). Since Gp fixes Q i and Q2, we have a Gp-action on k[[u]] and fc[[v]] making Equation A.7 Gp-equivariant, where Gp acts trivially on k. Since Gp is finite, there are positive integers m and n such that fc[[um]] = fc[[u]]Gp and fc[[t;n]] = fc[[w]]Gp. Since taking invariants preserves exactness ( |Gp| is invertible in k), we have that (k[[u, v]}\/(uvY)Gp is the kernel of the restriction of S to k[[uk]} \u00a9 k[[v% It follows that {k[[u,v]]\/(uv))Gp ~ k[[um, vn]}\/(umvn), which shows that 7r(P) is nodal since Cc,7r(P) \u2014 ( 7 I\" GC ,B) i r(p) \u2014 ^ B , P -It follows that E\\E\u00b0 is the set of points lying above C\\C\u00b0 and the number of points above Q G G \\ G\u00b0 is T ^ - T for any P G \u00a3 \\ S \u00b0 above Q. \u2022 Appendix A. Equivariant Riemann-Roch for Nodal Curves 66 A.9 Balanced action Let \u00a3 be a possibly singular G-curve over k. Suppose the G-action on E is admissible. Let Qi be the points on E lying above P G E. Then for any locally free G-O^.-module T of finte rank, XQ^JT are characters of the same group GQ{ = Gp as 77 is fully faithful. Def in i t ion 1.15. A n admissible G-action on a nodal curve E is ba lanced at a node P G E if XQLTQ- = XQ2 Q . A G-action is balanced if it is balanced at all nodes of E. A. 10 Equivariant Riemann-Roch for nodal curves T h e o r e m 1.16. Let G be a finite group and k be an algebraically closed field whose characteristic does not divide the order of G. Let E be a nodal projective curve over k with a generically free, admissible and balanced G-action, and let n : E \u2014> G = E\/G be the quotient. Let E\u00b0 be the smooth locus of E and let r\\ : E \u2014> E be the normalization. Let T be a locally free G-Os-module of finite rank. For each P G E let ep = \\Gp\\. Then 1 1 1 1 PEE\u00b0 j=o \/ 1 \\ 1 e p - l * E ' T = ( TGI D 6 G ? 7 * - F + X c ' \u00b0 c r k JJ X\"M ~ IG7 E E 3 \u2022 I ndg p (Xp^) \u2022 M l \/ I I P e E o j = 0 Proof. Note that the first statement follows easily from the second. Given a locally free G -0\u00a3-modu le T of finite rank, the canonical morphism T \u2014> ri^rfT is injective. Indeed the normalization n : E \u2014> E is a dominant morphism of algebraic varieties hence OE \u2014> V*\u00aeE is injective. Tensoring with T we get an injective morphism T \u2014\u2022 r)*0E \u00aeoE ? and by the projection formula we have r]*0E \u00aeoE J~ \u2014 V*1!*^- Thus we have an exact sequence of G-Og-modules 0 -> T -+ n+rfT -^K-+0 where 1Z is a G-Og-module supported at the singular locus of E. We have a long exact sequence of ^-representations of G 0 - HE(F) \u2014 K%{rt*T) > H\u00b0 (K) -- ElE(F) - KE(ri*f) HE(n) 0 where we used a G-equivariant isomorphism ^(77,77*F) ~ YP^rfF) which follows from j] being affine G-morphism by a Grothendieck spectral sequence applied to the functor FE\u00b0V* '\u2022 qCohG(E) \u2014\u2022 qCohG(\/c). Hence we have the following relation between characters XE,F \u2014 XErj,jr \u2014 XE,TZ- (A.8) Appendix A. Equivariant Riemann-Roch for Nodal Curves 67 The curve E is smooth and we can apply Proposition 1.10 XE,V* T={W\\ + XE\/G,OE\/G r k ^) Xk[G\\ QGE 3=0 By Lemma 1.11 and Lemma 1.14, we have XE\/G,Oe\/c = Xc,o6 = (*c,oc + \\C \\ C\u00b0\\) (A.9) = Xc,oc + E e p -1 1 PeE\\E\u00b0 Let P G E\\E\u00b0 and Q\\ and Q2 be the two points in E above P. Since the G-action on E is balanced, XQ2>Q- = X^ n _ and it follows that e Q l - l e<32-l E i \u2022 I ^ G Q i ( < * W ^ , n J + E J \u2022 I n d c Q 2 (^.^4^) j = 0 j=0 ep \u20141 ep \u20141 = E J \u2022 I n d c p (^4^) + E J \u2022 I n d c p (xpAxl^y) 3=1 j=i ep\u20141 ep\u20141 By Lemma 1.6, we have e p - l E J \u2022 I ndg p (Xps*Lns) + E (<* - \u2022 I n C (^4^) 3=1 3=1 e p - l ep - E I n d G p (XpfxL^-eP \u2022 E I ndg p (xP^QxU_) (A.10) j=i = ep \u2022 ^ E I n d G p (*^4i,n J - I n d G p ^ j (A.11) = eP \u2022 I E I n d G p (*P^<*v) - I n d g p Afp^ \\ V { E l r r f c Gp = ep \u2022 (Xp,H[d)\" *fc[G] - I n d G p XPtr) = eP \u2022 (rk(^) \u2022 X K [ G ] - I n d g p ^ ) where we used that every irreducible character of G p = GQ1 is of the form XQIQ_ for some 0 < j < ep \u2014 1 as the G-action on E is generically free. Thus using Equation A.9 to Appendix A. Equivariant Riemann-Roch for Nodal Curves 68 Equation A.11 and the fact that rj is an isomorphism over E\u00b0, we have XEtfT = ( | G J d e S ^ + Xc,oc R K ^ ) X m (A.12) j e Q - l - ] G | E \u00a3 j - I n d g Q ( * Q , ^ 4 , n J + |G7 E ep \u2022 rk(^) \u2022 ^ f c [ G ] ' ' PeE\\E\u00b0 E E i \u2022 l ^ o Q QeE\\r)-^(E\u00b0) 3=0 ( | G ] d e g v * : F + Xc,oc r k ^ *k[G] + IGT E e P \u2022 Indg\u201e # P i Jr. ' ' PeE\\E\u00b0 And finally the second statement of Theorem 1.16 follows from Equation A.8, Equa-tion A.12 and XE,n = xn\u00b0E(Ti) = TQ\\ E E P ' I n d G p *-P,^- ( A - 1 3 ) ' ' PEE\\E\u00b0 To show this, note that since TZ is a torsion sheaf H]j(7?.) = 0, and P\u20acE\\E\u00b0 P\u00a3E\\E\u00b0 But for Q \u20ac \u00a3 \\ \u00a3 \u00b0 we have 0 F p = I n d g Q F 0 . PeGQ And since Indg p Fp = Indg Q FQ for P e G \u2022 ( J , we have I n d g Q ^ = ^ \u00a3 I n d g p ^ 1 1 P6G-Q and Equation A . 13 follows. \u2022 69 Appendix B Galois Covers B . l Some Conventions and Notations Let X be a topological space. The fundamental groupoid TTI(X) is the groupoid whose objects are points of X and whose arrows are homotopy classes of paths whose source is the ending point and the target is the starting point. The composition of arrows in ni{X) is given by the following rules. Given two paths 7 and 6 such that the source of [7] is the target of [8] we define [y][5] as the homotopy class of the path 7 o 5(t) = 7(2*) for 0 < t < \\ and S(2t - 1) for \\ < t < 1. Given x and y G X, denote by wi(X,x,y) the set of arrows whose target is x and whose source is y, by TTI(X,X, \u2014) the set of arrows whose target is x, and by TTI(X,X) the group TTI(X,X,X). Given a G ni(X,x,y), denote by ca : n\\(X,y) \u2014> TV\\{X,X) the homomorphism (3 i-> a \/ ? a _ 1 . Suppose p : X \u2014> X is a covering. Let x G X and let x = p(x). Given a path 7 starting at x in X, denote by % the lift of 7 to the unique path in X starting at x. Given a = [7] G TTI(X,x,y) denote by x \u2022 a the element 7s(l) of E. This defines a right action of the fundamental groupoid ni(X) on X with anchor map p. Note that morphisms of covering spaces of X are 7ri(X)-equivariant. B.2 Pointed Principal C7-Bundles Let G be a finite group. Let p : E \u2014> C be a principal G-bundle. Note that E is a disjoint union of coverings of C. Hence the fundamental groupoid TTI(C) acts on E with anchor map p. Since G acts on E by isomorphisms of covering spaces we have the following. L e m m a 2.1. The left G-action and the right wi(C)-action on E commute, that is for x G E, g G G and a G TT\\ (C) such that the target of a is p(x) we have g \u2022 (x \u2022 a) = (g \u2022 x) \u2022 a. It follows that we have a well defined action A : G x TT1(C)opp x c E ^ E (g,a,x) h+g-x-a. It is easy to check that A is a transitive action if and only if C is path connected. A pair {E \u2014> C,x) consisting of a principal G-bundle E \u2014\u00bb C and a point x G E is called a po in ted p r i n c i p a l G-bundle . A morphism of pointed principal G-bundles is a morphism of principal G-bundles sending the marked point onto the marked point. L e m m a 2.2. Pointed principal G-bundles over path connected base space have no non-trivial automorphisms. Appendix B. Galois Covers 70 Proof. Let (p : E \u2014> C, x) be a pointed principal G-bundle with G path connected. Let 4> be an automorphism. Let y \u00a3 E, and because A is transitive there exist g G G and \/? G 7Ti(G,p(5),p(y)) such that g \u2022 x \u2022 (3 = y. Then since preserves x and is both G -equivariant and 7Ti(G)-equivariant, we have is trivial as y was arbitrary. \u2022 Let x G C and x G p~l{x). Given a G TTI{C, X), define 0(E,x)(a) as the unique element of G such that x-a = Q{E,x){a) \u2022 x-It follows from general properties of group actions that 0(E,X) 1S a group homomorphism w1(C,x)^G. L e m m a 2.3. Suppose C is path connected. Then E is path connected if and only if 9(E,X) is surjective. Proof. Suppose 9(E,X) is surjective. Let y G E and set y = p(y). Let a G TTI(C,X,y). Since 6(Etx) is surjective and A is transitive, there exists (3 G 7 r i ( G , X) such that 0(E,\u00a3)(P)'x'a = V-But 9(E,X)(P) \u2022 x \u2022 a = x \u2022 Pa which implies that x and y are in the same path connected component. The converse is trivial. \u2022 B.3 BG\\C Let C be a path connected, locally path connected, and locally simply connected space C. Suppose we are given a point x G G and a homomorphism 6 : 7 r i (G, X) \u2014\u2022 G . Let u : C \u2014> G be the universal covering space of G , and choose x' G u~1(x). Given a G 7Ti ( G , X), there is a unique isomorphism G such that ^(a)(a ; ' ) = x ' \u2022 a. This defines an action of Tti(C,x) on C . Let be the quotient space Ee = G x T 1 ( C x ) G of G x C by the left action of 7i\"i(G,:r) given by a \u2022 (g,y') = {g9(a~1),4>xi(a)(y')). Denote by [g,y'} the equivalence class of the pair (g,y') G G x C in Eg. Let \u2014> G be the map induced by u : C \u2014> C. We have a G-action on Eg induced by the G-action on the left factor of G x G , making Eg \u2014> C into a G-bundle. Clearly Eg \u2014> C is a principal G-bundle. The quotient map G x C \u2014> Eg is a morphism of covering spaces of G , hence it is 7Ti(G)-equivariant. It follows that the right 7Ti (G)-action on Eg is given by EgXcTT^C)^ Eg L e m m a 2.4. Let 6 : TT\\(C,X) \u2014> G 6e o homomorphism. There exists a pointed principal G-bundle pg : (Eg,xg) \u2014> (G, a;) SMC\/I i\/iai 0^Eg,xg) \u2014 0- Moreover the pair (Eg,xg) with the above property is unique up to a unique isomorphism of pointed principal G-bundles. Appendix B. Galois Covers 71 Proof. Let Eg be the bundle constructed above and let xg = [id,x'}. Let P G iri(C,x), then we have 0(Eg,ie)(P) ' x0 = xe \u2022 P = [id, x \u2022 P] = [id,4>X'{P){x')] = [9{P)1x'} = 0(P) \u2022 xg and it follows that 9{Eg:Xg)(P) = 0(P). Let p : (F, y) \u2014> (C, a;) be a pointed principal G-bundle such that 0(F,y) \u2014 B y the property of universal coverings, there exists a unique morphism : (U, x') \u2014> (F, y) such that p o = u. Define : (G x U, {id,x')) -> {F,y) by $\"{g,y') = g \u2022 T o P r o v e that descends to a morphism <\u00a3 : (Eg,xg) \u2014> (F, j\/) we must show that $\" is constant on the 7ri(G,x)-orbits. Let a G 7Ti(C,a;), (5,2\/') G G x U and \/3 G iri(C,x,u(y')). We have = ^ ( a _ 1 ) \u2022$'(^(\")(2\/')) = gO{a-x) \u2022&(4,x,(a)(x'-P)) = g9{a-x) \u2022 '( (F, y). It is clearly G-equivariant, and thus an isomorphism of pointed principal G-bundles. It is unique as pointed principal G-bundles have no nontrivial automorphisms (Lemma 2.2). \u2022 Let G ^ 1 ^ ) denote the set Hom(7ri(G,x),G) of homomorphisms from wi(C,x) to G. The group G acts on the left of G^0'^ by conjugation on the target. L e m m a 2.5. Given two homomorphisms 9\\ and 62 : iri(C,x) \u2014\u2022 G, there exists an iso-morphisms of principal G-bundles Eg1 \u2014> Eg1 if and only if 9\\ and 92 lie in the same G-orbit in G ^ 0 ^ . More precisely, suppose h G G is such that 92 = Ch \u00b0 Oi, then there exists a unique isomorphism of principal G-bundles Qgltg2(h) : Eg1 \u2014> Eg2 such that \u00a7gue2{h){xei) h 1 \u2022 xg2. Proof. Suppose first that 92 = Cho9\\ for some h G G . We know that for i = 1 or 2, Egt is the quotient of G x U by the right action of ni(C,x) given by {g, y')-a = (g9i(a), (f>xi{a~l){y')). Denote by [g,y% the equivalence class of (g,y') in Egt. We define $g1:g2(h) : Eg1 \u2014> Eg2 Appendix B. Galois Covers 72 by sending the equivalence class [x,(a-1)(y% = {gh-1e2(a), x,(a-1)(y% = {gh-\\y'h = * e i , f c W ( [ f f , S \/ ' ] i ) -Clearly $>g1;e2(h) is an isomorphism of principal G-bundles, sending xg1 = [id,x']\\ onto [h~l,x'}2 = h~l \u2022 xg2. Again $gltg2(h) is unique such isomorphism as pointed principal G-bundles have no nontrivial automorphisms. Conversely suppose that we have an isomorphism of principal G-bundles $ : E$1 \u2014* Eg2. Then let h e G be the unique element such that ^(xg^) = hT1 \u2022 XQ2. Then 92{p) \u2022 xg2 = xg2 \u2022 P = h-*(x01)-p = h \u2022 $(x6l \u2022 P) = h-Q{di{p)-xei) = h9l{p)-${xei) = ch(6i(P)) \u2022 * BG\\C that sends a homomorphism 9 : TT\\(C,X) \u2014> G onto the principal G-bundle Eg, and sends an arrow g : # i \u2014 > 62 onto the isomorphism $e^,e2{g) '\u2022 Eg1 \u2014> Eg2. Proof. We first check that E is a well defined functor. Let 0j e G 7 1 \" 1 ^ ) for i = 1,2,3 and gi \u00a3 G for i = 1,2 such that 02 = cgi o d\\ and #3 = cg2 o 92. Then by Lemma 2.5, S ( f f 2 : # 2 \u2014* #3) 0 2 ( 5 1 : 9\\ \u2014> 92) is the unique isomorphism Eg1 \u2014* \u00a3#3 sending a;^ o n t o gi1g21 \u2022 ^e 3, that is E(g2gi : 9\\ \u2014> #3). Thus 5 is well defined. We now show that S is fully faithfull. Let 9\\ and # 2 \u00a3 GT1^C'X\\ We want to show that 5 : H o mG ( G ' r i ( c ^ ) ) ( 6 , i ' 6 , 2 ) HomBG\\c(Eei,Ee2) g ^$eug2{g) Appendix B. Galois Covers 73 is a bijection. It is injective as g is the unique element of G such that ^gltg2(g)(xg1) = g~~lxg2. It is surjective by Lemma 2.5. To show that E is essentially surjective, let p : E \u2014> C be a principal G-bundle, choose x G p^1x, and note that by the Lemma 2.4, Eg{E - } is isomorphic to E. \u2022 C o r o l l a r y 2.7. \/ \/ A u t ^ - E g ) denotes the G-equivariant automorphisms of Eg overC, then there is an isomorphism : Aut\u00b0.(\u00a3#) \u2014\u2022 Z G ( 0 ( 7 T I ( G , X))), such that for 4> \u00a3 A u t g ( \u00a3 e ) = * e ( ^ ) _ 1 -X0. Proof. Since S is a fully faithfull it induces an isomorphisms of groups H o mG ( G * i ( c , * ) ) ( M ) -> H o m B G | c ( \u00a3 ; e ) . The left side is just the centralizer ZC(9(TTI(C, X ) ) ) and the right side is the automorphism group AutQ(Eg). Let (f> G Aut^,(Eg), and let 5 G G be the element mapping onto . Then by Lemma 2.5 C\u00b0, there exists a unique branched covering E \u2014> C and a map E \u2014> E such that E E C\u00b0 \u00bb- C is a pull back diagram. If E \u2014> C\u00b0 is a G-bundle then E \u2014> C has a unique structure of a G-bundle such that the above diagram is a pull back diagram of G-bundles (Forster 8.4 and 8.5). For each 1 < i < n choose a path di in G \u00b0 U {x^} from \u00a3 0 to x^, such that di passes through Xi only once. Given XQ G p~l(xo) we define \u00a30 \u2022 di G E as l i m c _ i XQ \u2022 d\\ where d\\ is the path in G \u00b0 given by 11\u2014> d,(ct). L e m m a 2.8. For each 1 < i < n, there exists an cti G wi(C0,xo) such that for every finite group G and every homomorphism 9 : 7 T i ( C \u00b0 , xo) \u2014> G , the element 9(ai) fixes xg-di G Eg and acts on TXe.(iiEg by multiplication by exp(27rv\/\u20141\/fcj) for ki = \\ Stabc(xg \u2022 di)\\. Proof. F i x 1 < i < n. Let cj>: D2 \u2014> G \u00b0 U {a;,} be an embedding of the closed disk D2 CC of radius 1, with image D, sending 0 to Xi such that XQ 0 (exp(27rv \/-li) \u2022 4>~l(di(c))). Define Q j G 7Ti(C\u00b0, xo) as the homotopy class of the path d\\ o (5 o I claim that Qj depends only on di. Let 0' : D2 \u2014> C\u00b0 U {ZJ} be another embedding of the closed disk D2 C C of radius 1, with image D'. Let c' G [0,1] and <5' : [0,1] -> dD' be defined as above with D replaced by D'. Assume that D' C D. Let dcc : [0,1] \u2014> D \\ {a:,} be the path given by 1>\u2014> di(t(c' \u2014 c) + c) (the path along from di(c) to di(c')). Then it is easy to construct a homotopy between dcc o 5' o (d\u00a3 ) _ 1 and 5 in D \\ {x^, thus [dio5'o{diri} = [5] and it follows that [d$\u00b0so {dir1} = [di o 4 o y o [dir' o (dir1} = [dfos'o(dfr1} which shows that a\u00ab doesn't depend on the choice of D' C D. Let G be a finite group and 9 : iri(C\u00b0,xo) \u2014> G a homomorphism. Choose (p : D2 \u2014\u00bb G\u00b0 U {a i j} , .D, c and <5 as above. Then \u00a3 # | . D is a disjoint union of disks and there exists an embedding

~~\u2014> zk for k = \\ Stabc^e)! (by Forster 5.10). Hence the element [6] G TTI(D \\ {xi}, di(x)) fixes xg \u2022 di and acts on T^.^Eg by'multiplication by exp(27T\\\/\u2014T\/k), and so does on. \u2022 For each 1 < i < n choose a, G \/KI(C\u00b0,XO) as in the Lemma. Let fTj < ai >\u2014< oti > x \u2022 \u2022 \u2022 x < an > act on the right of Gn x G^c\u00b0'xo\\ by G n x Q^(C,X0) x JJ < Q i > ^ G \" X Gvi(c\u00b0>xo) i Let G act on the left of G n x G 7 ^ 0 - * 0 ) by G x G\u2122 x <57ri(c'0>:i:o) \u2022 G 7 1 x ( j^ i lG 0 , ^ ) (g,9i,9) ' ^ (g%9~l,Cg o 9). Note that the two actions commute. Appendix B. Galois Covers 75 Proposition 2.9. Let MGCxi Xn^ be the category whose objects are tuples (p : E \u2014> C, x\\,... ,xn) where p : E \u2014> C is a G-bundle that restricts to a principal G-bundle on C\u00b0 and Xi G p~l(xi) are marked points, and the arrows are isomorphisms of G-bundles preserving the marked points. Denote by A\/^ G x i x ^ the transformation groupoid of the G-action on the quotient (Gn x G7Tl(-c\u00b0,x\u00b0^)\/(Yli < on >). Then we have an equivalence of categories & G : \u2022 A \/ ( C , x 1 , . . . , x n ) - > Mfc,Xl,...,xn) sending an object [gi,0] G (Gn x G 7 r i(G\u00b0' : ! ; o))\/(rTj < cti >) onto (Eg \u2014> C,gi-xg-di) and an arrow g \u20ac G from [gi,9] to [hi,p] onto the unique morphism of G-bundles >P(g) : Eg \u2014> Ep extending $gtP(g). Proof. We start by showing that 9 G is well defined. Let (fcj) E Z\u2122. Then giff(a^) -xg-di = gi \u2022 xg \u2022 di by Lemma 2.8. So \u00a9 G is well defined on the objects. Given g G G , we need to check that QgtP(g) preserves the marked points. Note that hi = gig-1 and p = cg o 9. Thus we have \u00aee,cgoe(g)(9i \u2022 xg \u2022 di) = gt \u2022 \u00a7etCgoe{g){xe) \u2022 di = gig\"1 \u2022 xCg0g \u2022 di \u2014 hi \u2022 Xp ' di. We now show the essential surjectivity. Let (p : E \u2014\u00bb G, ii) be an object of MG(x\\, ... ,xn and choose Xo G p~1(xo). Let E\u00b0 be the restriction of E to G\u00b0. Let 6 \u2014 9(B\u00b0,x0) a n d choose gi G G such that gi - XQ- di = Let us check that QG[gi,8] = (Eg \u2014> C,gi \u2022 xg \u2022 di) is isomorphic to (E \u2014> C,Xi) as pointed G-bundles. By Lemma 2.4, we have an isomorphism $ : Eg \u2014> E\u00b0 sending xg onto x$, inducing an isomorphism $ : Eg \u2014> E. We need to check that $ preserves the marked points, which follows from the following computation $(9i \u2022 xe \u2022 di) = gt \u2022 $(xe) \u2022 di = 9i-xo-di We show that 0 G is fully faithfull as follows. Let 9 and p : TTI(C\u00b0,XO) \u2014> G be two homomorphisms, and let gi, hi G G . Given $ : (Eg,gi \u2022 xg \u2022 di) \u2014> (Ep, hi \u2022 xp- di), there is a unique g G G such that $(gtP)(g) is the restriction of~~ZC(9(K\\(C\u00b0 ,x0))) restricts to an isomorphism Ant%(Eg,gi -xg-du-.^gn-xg-dn) -\u00bb Zg(9{TT1{C\u00b0 , x0)))n < 6(ai) > n... n < 9(an) > where < 9(on) > denotes the subgroup of G generated by 6(cti). Proof. A n element g e G gives an arrow from [gi, 9] to itself if and only if cg o 9 = 9 that is g \u20ac ZC(9(TTI(C\u00b0,XQ))), and ftg-1 = ft modulo Z that is 3 \u20ac< 9(oti) >. \u2022 B. 5 Inflation Suppose that i f is a subgroup of G or more generally that there is a homomorphism i f \u2014> G. There is a functor G xH \u2014 from the category of if-spaces to the category of G-spaces which takes an if-space X and sends it to the quotient G x H X of the product G x X by the action (g,a;) \u2022 h \u00bb-> (g \u2022 h,h _ 1 a;) . The space G Xfj X has a natural G-action induced by the multiplication on G . This construction is functorial. Note that there is a natural if-morphism GxH : X \u2014> G x# X induced by the if-morphism H x X \u2014> G x X and the identification X ~ H XJJ X. Denote by 4 the inclusion i f \u2014> G . For simplicity write A4H := M.\u00a5N and A 4 G := A4\u00a3, \u201e. v We will construct a funtor ML : MH \u2014> A 7 ! 0 called the inflation. Given an object e := (p : E -* C,x\\,... ,xn) of A4^Cxi Xny let A ^ e ) := (q : F \u2014> C, yi,... ,yn) be the object of A - l 0 - , \u201e N constructed as follows: let F:=GxHE and for each i , let Vi \u2022= G xH (\u00b1i). We use the convention that two functors of groupoids are said to commute if they are naturally isomorphic. Proposition 2.11. Suppose that 1 : H \u2014* G is an injective homomorphism of groups. Then we have a commutative diagram of groupoids AfH \u2014\u2014*\u2022 j\\4H A\/\"1 AfG \u2014 ^ - \u00bb - M G where A \/ 1 is the functor AfH \u2014> AfG induced by the inclusion i f \" X if 7 ri( c' 0> : ro) Qn x QTTI(C\u00b0,X0) Appendix B. Galois Covers 77 Proof. We have to construct a natural isomorphism r : 0 G ojV\"' ~ A4l o QH. Let [hi, 9} \u00a3 (Hn \u2022^H^c\u00b0'x\u00b0y)\/(X[i < cti >), and let 9' = i o 9. Taking the quotient of inclusion H x C\u00b0 \u2014\u2022 G x G\u00b0 by the action of 7 r i ( G \u00b0 , X o ) , we get if-equivariant morphism Eg \u2014> which induces a G-equivariant isomorphism T{9) : G xH Eg \u2014> Eg\/ that sends G x# (xg) onto aV, and thus preserves the markings. This is the required natural isomorphism. \u2022 C o r o l l a r y 2.12. The functor : MH \u2014\u2022 A 4 G is faithful and ifn>lit is fully faithful. Proof. By Corollary 2.10 and Proposition 2.11, the inflation AAl induces a homomorphism of groups Aut G (Eg,hi -xg \u2022 di) \u2022\u2014: >\u2022 Aut G (\u00a3 t o e, t-(hi) \u2022 xLOg \u2022 di) ZH(9(7T1(Co,x0))) n (6(ai)) n . . . n (9(an)) ^ Z G ( t o ^(7n(G\u00b0, x 0 ))) n (L o 0(ai)) n . . . n (t o 0(a\u201e)> where the bottom arrow is an inclusion and hence A4L is faithfull. It is also an isomorphism if n > 1, and to the full faithfulness we only need to show that two objects in A4H are isomorphic if and only if their images in MG are isomorphic. B y Proposition 2.11 it suffices to show this property for AfH and AfH. Let [hi, 6} \u00a3 (Hn x Hv^c\u00b0'x\u00b0^>) \/ < on >), and suppose that for g \u00a3 G , we have [hi \u2022 g~l,cg o 9] \u00a3 (if\u2122 x # 7 r i ( G \u00b0> x \u00b0 ) ) \/ ( rL j < a\u00bb >). Then since n > 1, we have cg o #(OJI) G i f , which implies that \/ i i \u2022 g~x \u00a3 H and it follows that g\u00a3H. \u2022 B.6 Monodromies Consider the map efc,Xl,...,Xn) \u2022\u2022 (Gn x G^G\u00b0^))\/(n )^Gn . It is invariant under the left G action, and thus induces a functor e (C ,x i , . . . , x\u201e ) \u2022 ^ ( C x i , . . . , ^ ) ~ ^ U where G is considered as a category associated to the set G . Given an n-tuple (i\/\\,..., vn) \u00a3 Gn, define Af^Xl x^(vi,... ,un) as the full subcategory of A ^ G X i x ^ with objects in the fiber of efCtXli\u201e.tXn) above (J\/I, . . . , un). Note that A ^ G X i Xn){vu \u2022 \u2022 \u2022, vn) is the transformation groupoid of the G-action on the set {efcxi x -))~1(i\/i, \u2022 \u2022 \u2022, Vn)-Def in i t i on 2.13. Given a G-bundle p : E \u2014> C of Riemann surfaces such that it is a principal G-bundle away from some x \u00a3 C. Given x \u00a3 p~x{x) we define the m o n o d r o m y of E \u2014y C at x as the unique element vx of G that fixes x and acts on the tangent space TXE by multiplication by eyLp(2ir\\f^l\/k) where k = | Stab(j(x)|. Appendix B. Galois Covers 78 Proposition 2.14. Given an n-tuple (v\\,...,vn) \u20ac Gn, the equivalence of categories in Proposition 2.9 restricts to an equivalence of categories JV (C , A ! L , . . . , * B )K \u2022\u2022\u2022,\"n)-+ M f C t X l _ X n ) ( v U - \u2022 \u2022 , Vn) where M.GCxi^(v\\,..., un) is the full subcategory of MGCxi x^ whose objects are the marked G-bundles (E \u2014> C, x\\,... ,xn) whose monodromy at Proof. Given [git 9) e (efcxi z , , ) ) - 1 ^ 1 ' \u2022 \u2022 \u2022 > ^ follows from Lemma 2.8 that the mon-odromy of xo \u2022 di is 9(cn). Hence the monodromy of gi \u2022 XQ \u2022 di is gi9(cti)g^1. The result then follows. \u2022 79 Bibliography [ACV03] D . Abramovich, A . Corti, and A . Vistoli . Twisted bundles and admissible covers. Comm. 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