UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Finite groups of low essential dimension Duncan, Alexander Rhys 2011

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2011_fall_duncan_alexander.pdf [ 356.7kB ]
Metadata
JSON: 24-1.0071872.json
JSON-LD: 24-1.0071872-ld.json
RDF/XML (Pretty): 24-1.0071872-rdf.xml
RDF/JSON: 24-1.0071872-rdf.json
Turtle: 24-1.0071872-turtle.txt
N-Triples: 24-1.0071872-rdf-ntriples.txt
Original Record: 24-1.0071872-source.json
Full Text
24-1.0071872-fulltext.txt
Citation
24-1.0071872.ris

Full Text

Finite Groups of Low Essential Dimension by Alexander Rhys Duncan  H.B.Sc., The University of Toronto, 2005 M.Sc., The University of British Columbia, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) June 2011 c Alexander Rhys Duncan 2011  Abstract Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.  ii  Preface Portions of Chapters 1, 2, 3, 4, 5, 6 and 8 appear in: Alexander Duncan. Finite groups of essential dimension 2, 2010. Accepted in Comment. Math. Helv. A version of Section 6.1 appears in: Alexander Duncan. Finite groups of essential dimension 2, 2009. arXiv:0912.1644v1 [math.AG]. Portions of Chapters 1, 7 and 8 appear in: Alexander Duncan. Essential dimensions of A7 and S7 . Math. Res. Lett., 17(2):263–266, 2010  iii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  vi  Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Symbols  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . Dedication  xi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Essential Dimension via Birational Geometry . . . . . . . . .  2  1.2  Finite Groups of Essential Dimension 2 . . . . . . . . . . . .  3  1.3  Essential Dimension of A7 and S7  . . . . . . . . . . . . . . .  5  1.4  Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  2.1  Versal Varieties  2.2  Minimal Rational Surfaces  . . . . . . . . . . . . . . . . . . .  11  2.3  Polyhedral Groups . . . . . . . . . . . . . . . . . . . . . . . .  11  3 Versal Actions on Toric Varieties . . . . . . . . . . . . . . . .  13  3.1  . . . . . . . . . . . . . . . . . . . . . . . . .  Cox Rings and Universal Torsors . . . . . . . . . . . . . . . .  7  13 iv  3.2  Monomial Actions . . . . . . . . . . . . . . . . . . . . . . . .  18  4 Del Pezzo Surfaces of Degree ≥ 5 . . . . . . . . . . . . . . . .  22  4.1  Monomial Actions on Toric Surfaces . . . . . . . . . . . . . .  23  4.2  Versal Actions on the Four Surfaces . . . . . . . . . . . . . .  30  5 Conic Bundle Structures  . . . . . . . . . . . . . . . . . . . . .  34  6 Del Pezzo Surfaces of Degree ≤ 4 . . . . . . . . . . . . . . . .  41  6.1  Alternative Proofs . . . . . . . . . . . . . . . . . . . . . . . .  43  6.1.1  Del Pezzo Surfaces of Degree 4 . . . . . . . . . . . . .  43  6.1.2  Del Pezzo Surfaces of Degree 3 . . . . . . . . . . . . .  44  7 Essential Dimensions of A7 and S7 8 Conclusion  . . . . . . . . . . . . . . .  52  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  8.1  Applications  . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  8.2  Future Research . . . . . . . . . . . . . . . . . . . . . . . . .  57  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  60  v  List of Tables 4.1  Conjugacy classes of non-trivial finite subgroups of GL2 (Z) .  24  4.2  Versality conditions for monomial actions on surfaces . . . . .  31  vi  List of Figures 4.1 4.2  Lattice of finite subgroups in GL2 (Z) . . . . . . . . . . . . . . Standard fans for  P2 ,  P1  ×  P1  and DP6 . . . . . . . . . . . . .  24 25  vii  List of Symbols k  base field  Q  rational numbers  R  real numbers  C  complex numbers  Fp  finite field of p elements  k(x1 , . . . , xn ) field extension of k generated by x1 , . . . , xn ζn  primitive nth root of unity  det(M )  determinant of matrix M  Tr(M )  trace of matrix M  x  greatest integer less than x  Hom(A, B)  set of homomorphisms from A to B  A  A and B are isomorphic  B  ed(G)  essential dimension of the group G over C  edk (G)  essential dimension of the group G over k  covdim(G)  covariant dimension of group G  Groups G  finite group  Cn  cyclic group of order n  D2n  dihedral group of order 2n  An  alternating group on n letters  Sn  symmetric group on n letters  k×  multiplicative group of field k  GLn (R)  general linear group of dimension n over the ring R  SLn (R)  special linear group of dimension n over the ring R viii  PGLn (R)  projective general linear group of dimension n over the ring R  PSLn (R)  projective special linear group of dimension n over the ring R  Gp  Sylow p-subgroup of finite group G  Gi  subgroup of GL2 (Z) (see Table 4.1)  rank(A)  rank of abelian group A  Z(G)  centre of group G  A  semidirect product of A and B (here A is normal)  B  g1 , . . . , gn  group generated by elements g1 , . . . , gn  W (R)  Weyl group of root system R  Aut(X)  automorphisms of X  StabG (p)  stabilizer of p in group G  XG  points of X fixed by the group G  Xg  points of X fixed by the group g  X/G  quotient of X by G  RG  invariant ring of R by G  IndG Hσ 2 H (G, K)  representation of G induced from representation σ of H  ResG H  restriction map from G to H in group cohomology  second group cohomology group of G in K  Algebraic Geometry Pn  projective space of dimension n  Cn  affine complex n-space  DP5  the del Pezzo surface of degree 5  DP6  the del Pezzo surface of degree 6  Fn  nth Hirzebruch surface  dim(X)  dimension of variety X  k(X)  function field of variety X  X  Y  rational map from X to Y  Pic(X)  Picard group of variety X  Spec(R)  affine scheme associated to ring R  OX  structure sheaf of variety X  OX (n)  n-twisted structure sheaf of variety X  ix  P(F)  projective space bundle associated to F  KX  canonical sheaf of variety X  Γ(X, F)  global sections of sheaf F on X  H 1 (X, A)  first ´etale cohomology group of X in A  Toric Varieties ∆  fan in a lattice  ∆X  fan associated to toric variety X  X(∆)  toric variety associated to the fan ∆  ∆(1)  set of rays in the fan ∆  DivT (X)  group of T -invariant divisors of toric variety X  Aut(N, ∆)  automorphisms of lattice N preserving the fan ∆  AutT (X)  T -stable automorphisms of toric variety X  ωT  canonical morphism from AutT (X) to Aut(N, ∆)  Cox(X)  Cox ring of toric variety X  Aut(X)  preimage of Aut(X) in Aut(Cox(X))  Conic Bundles X  total space of conic bundle  B  base space of conic bundle  φ  the map X → B  π  map G → PGL2 (C) induced by action of G on B  GK  kernel of π  GB  image of π  Σ  set of points on B whose fibres are singular  pi  point in Σ  R  set of components of singular fibres  Ri , R i  components of φ−1 (pi )  Aut(R)  set of permutations of R  ξ  natural map G → Aut(R)  G0  group ker(ξ) ∩ GK  x  Acknowledgements I thank my advisor Zinovy Reichstein for introducing me to essential dimension and many of the geometric techniques used in this thesis. His career and academic advice were invaluable throughout my time at UBC. I thank my committee members Patrick Brosnan, for several helpful discussions, and Kalle Karu, for introducing me to toric varieties and to Cox rings. In addition, I would like to acknowledge Roland L¨otscher, Aurel Meyer, Yuri Prokhorov, Jean-Pierre Serre and two anonymous referees for useful comments and correspondence.  xi  Dedication To my parents  xii  Chapter 1  Introduction Loosely speaking, the essential dimension of an algebraic object is the minimal number of parameters needed to describe it. It was introduced for finite groups by Buhler and Reichstein in [12]; for algebraic groups by Reichstein in [49]; for algebraic stacks by Brosnan, Reichstein and Vistoli in [10]; and for functors by Merkurjev in [7]. Essential dimension is of great interest in Galois cohomology, linear algebraic groups, and central simple algebras; has applications to Galois theory and the Noether problem; and is related to several open problems in algebra such as Albert’s cyclicity conjecture, Hilbert’s 13th problem, and Serre’s Conjecture II. The essential dimension of the symmetric group on n letters, ed(Sn ), has connections to the simplification of polynomials via Tschirnhaus transformations. In Buhler and Reichstein’s original paper [12], their main interest was to determine how much a “general polynomial of degree n” can be simplified in this manner. This may be viewed as a modern continuation of classical results of Hermite, Joubert, Klein and Hilbert. The essential dimension of general finite groups is of interest in inverse Galois theory. Here one wants to construct polynomials over a field k with a given Galois group G. Ideally, one wants polynomials that parametrize all fields extensions with that group: the so-called generic polynomials (see [36] and [34]). The essential dimension of G is a lower bound for the generic dimension of G: the minimal number of parameters possible for a generic polynomial. This thesis will mainly consider finite groups of low essential dimension using birational geometry; we outline our approach in Section 1.1. The main results of this thesis are a classification of all finite groups of essential di1  mension 2 (see Section 1.2), and a determination of the essential dimensions of the alternating group, A7 , and the symmetric group, S7 (see Section 1.3).  1.1  Essential Dimension via Birational Geometry  Our study of essential dimension uses the concept of a versal G-variety (defined in Chapter 2). These are simply models of the versal torsors seen in Galois cohomology. The key fact we use is that if G is a finite group of essential dimension n then there exists a versal G-variety of dimension n. This suggests an approach for classifying groups of essential dimension n: consider each faithful G-variety of dimension n and determine whether it is versal. Of course, enumerating all possible G-varieties of a given dimension is completely impractical. We will see that it suffices to consider considerably smaller families. For dimension 2, we need only consider minimal rational G-surfaces. The minimal rational G-surfaces were classified by Manin [45] and Iskovskikh [33] building on work by Enriques: they either possess conic bundle structures or they are del Pezzo surfaces. The use of the Enriques-Manin-Iskovskikh classification for computing essential dimension was pioneered by Serre in his proof that edk (A6 ) = 3 [58, Proposition 3.6]. Independently, Tokunaga [60] has also investigated versal rational surfaces. The dichotomy into conic bundle structures and del Pezzo surfaces is too coarse to easily identify exactly which groups occur. Our current work was inspired by Dolgachev and Iskovskikh’s [25] finer classification of such groups. Their goal was to classify conjugacy classes of finite subgroups of the Cremona group of rank 2 (the group of birational automorphisms of a rational surface). This problem has a long history. The first classification was due to Kantor; an exposition of his results (with some corrections) can be found in Wiman [61]. Unfortunately, this early classification had several errors, and the conjugacy issue was not addressed. More recent work on this problem include [4], [21], [62], [5] and [8]. For higher dimensions, we are interested in unirational G-varieties. Unlike surfaces, we do not have powerful classification theorems. However, 2  Prokhorov recently used ideas from the minimal model program to classify finite simple groups which act faithfully on rationally connected threefolds [48]. This work is crucial for our computation of the essential dimensions of A7 and S7 .  1.2  Finite Groups of Essential Dimension 2  Let k be an algebraically closed field of characteristic 0. We use the notation D2n to denote the dihedral group of order 2n. Finite groups of essential dimension 1 were classified by Buhler and Reichstein in their original paper; they are either cyclic or isomorphic to D2n where n is odd. There is a classification for infinite base fields by Ledet [42] (see also Remark 3.2), and for arbitrary base fields by Chu, Hu, Kang and Zhang [14]. We review what is known about groups G of essential dimension 2. If G contains an abelian subgroup A then rank(A) ≤ 2. The Sylow p-subgroups Gp of G can be described using the Karpenko-Merkurjev theorem [35]: Gp must be abelian for all p odd, and groups G2 must be of a very special form (see [47, Theorems 1.2 and 1.3]). Any subgroup of GL2 (k) or S5 has essential dimension ≤ 2. Finite groups of essential dimension 2 with non-trivial centres were classified (implicitly) by Kraft, L¨otscher and Schwarz (see [40] and [39]). They show that a finite group with a non-trivial centre has essential dimension ≤ 2 if and only if it can be embedded in GL2 (k). Their main interest was in covariant dimension, a “regular” analog of essential dimension. See also [50] and [44]. Recall that the automorphism group of the algebraic group (k × )n is isomorphic to GLn (Z). Our main theorem is as follows: Theorem 1.1. Let T = (k × )2 be a 2-dimensional torus. If G is a finite group of essential dimension 2 then G is isomorphic to a subgroup of one of the following groups: 1. GL2 (k), the general linear group of degree 2,  3  2. T  G1 with |G ∩ T | coprime to 2 and 3  G1 = 3. T  01 10  D12 ,  −1 0 0 1  ,  01 10  D8 ,  G3 with |G ∩ T | coprime to 3  G3 = 5. T  ,  G2 with |G ∩ T | coprime to 2  G2 = 4. T  1 −1 1 0  0 −1 1 −1  ,  0 −1 −1 0  S3 ,  G4 with |G ∩ T | coprime to 3  G4 =  0 −1 1 −1  ,  01 10  S3 ,  6. PSL2 (F7 ), the simple group of order 168, 7. S5 , the symmetric group on 5 letters. Furthermore, any finite subgroup of these groups has essential dimension ≤ 2. The proof of Theorem 1.1 breaks into two mostly independent pieces. We show that it suffices to consider only four surfaces: Theorem 1.2. If G is a finite group of essential dimension 2 then G has a versal action on one of the following: the projective plane P2 , the product of projective lines P1 × P1 , or a del Pezzo surface of degree 5 or 6. Then, we show that the groups with versal actions on these four surfaces are those listed in Theorem 4.5 above. We also mention some intermediate results that we feel are of independent interest. Many interesting versal varieties are toric. In order to classify versal actions on toric varieties, we develop techniques that apply to smooth complete toric varieties of arbitrary dimension. Our major tools are the theory of Cox rings [19] and universal torsors [16]. The main result on toric varieties is as follows (Theorem 3.2): a faithful G-action on a complete nonsingular toric variety is versal if and only if it lifts to an action on the variety of the associated Cox ring. This result has some important corollaries. First, if a complete nonsingular toric variety has a G-fixed point then it is versal (Corollary 3.5). 4  Second, a complete non-singular toric variety is G-versal if and only if it is Gp -versal for all of its p-subgroups (Corollary 3.6). This second corollary is instrumental in our proof of Theorem 4.5.  1.3  Essential Dimension of A7 and S7  The essential dimensions of the alternating groups, An , and the symmetric groups, Sn , are of special interest because they relate to classical questions of simplifying degree n polynomials via Tschirnhaus transformations. In particular, the degree 7 case features prominently in algebraic variants of Hilbert’s 13th problem. In this language, several results for small n were established by Hermite, Joubert and Klein in the 1800s. For more information, see Chapter 8. The values of edk (Sn ) and edk (An ) are known for all n ≤ 6; see [12] and [58, Proposition 3.6]. The second main result of this thesis is the following: Theorem 1.3. Let k be a field of characteristic 0 (not necessarily algebraically closed). Then edk (A7 ) = edk (S7 ) = 4. The proof relies on recent work of Prokhorov [48] on the classification of rationally connected threefolds with faithful actions of non-abelian simple groups.  1.4  Overview  The rest of the thesis is structured as follows. In Chapter 2, we recall basic facts about versal varieties, essential dimension and the Enriques-Manin-Iskovskikh classification. In Chapter 3, we develop tools for determining when a toric G-variety is versal. Chapters 4 through 6 are devoted to finite groups of essential dimension 2. In Chapter 4, we determine precisely which groups act versally on the four surfaces of Theorem 1.2. In Chapter 5, we show that all groups acting versally on conic bundle structures already act versally on the four surfaces.  5  In Chapter 6, we show the same for the del Pezzo surfaces. This proves Theorem 1.2 and, thus, Theorem 1.1. Chapter 7 is a proof of Theorem 1.3. Finally, Chapter 8 discusses applications of these results, and directions for future research.  6  Chapter 2  Preliminaries 2.1  Versal Varieties  In this thesis, a variety is an integral separated scheme of finite type over a field k. We assume all actions and maps are defined over k. Let G be a finite group. A faithful G-variety is a variety with a faithful Gaction. A compression is a G-equivariant dominant rational map of faithful G-varieties. Given a faithful G-variety X over k, the essential dimension of X, denoted edk (X), is the minimum dimension of Y over all compressions X  Y where Y is a faithful G-variety over k. The essential dimension of  G, denoted edk (G), is the maximum of edk (X) over all faithful G-varieties X over k. Unless explicitly stated otherwise, we will always take k = C. Our main results apply for more general fields, but it suffices to consider C for Theorem 1.1 by the following lemma: Lemma 2.1. Suppose G is a finite group and k is a field of characteristic 0. (a) If k /k is a field extension then edk (G) ≥ edk (G). (b) If k is algebraically closed then edk (G) = edC (G). Proof. Part (a) is well-known; a proof can be found in [7, Proposition 1.5]. Part (b) is just [10, Proposition 2.14(1)] since k and C both contain an algebraic closure of Q. A similar argument will be used in the proof of Theorem 1.3. For the remainder of the thesis, we write ed(−) to mean edC (−) without risk of ambiguity. 7  If H is a subgroup of G then ed(H) ≤ ed(G); a similar inequality fails for quotient groups [47, Theorem 1.5]. The essential dimensions of the symmetric groups, Sn , and alternating groups, An , are known for n ≤ 6 and bounds exist for higher n (see Chapter 7). It is a deep result of Karpenko and Merkurjev [35] that the essential dimension of a p-group is the minimal dimension of a faithful linear representation. Definition 2.1. A G-variety X is G-versal (or just versal ) if it is faithful and, for any faithful G-variety Y and any non-empty G-invariant open subset U of X, there exists a G-equivariant rational map f : Y  U . We say an  action of G is versal, or that G acts versally, if the corresponding G-variety is versal. Note that the versal property is a birational invariant: it is preserved by equivariant birational equivalence. Remark 2.1. In algebraic geometry, one often studies the category of Gtorsors over schemes by studying the classifying stack BG. Indeed, one can obtain any G-torsor by pulling back along a unique morphism to BG. One may then view the classifying stack as a “universal G-torsor”. One would like to do something similar in Galois cohomology with G-torsors over fields. However, the category of field extensions of k is not as flexible as that of schemes. Morally, a “versal torsor” (see [7, Definition 6.3] or [27, Definition 5.1]) is a G-torsor over a field from which any other G-torsor can be obtained by pulling-back along a rational map. For this to make sense, one must “thicken” the torsor by constructing a variety with same generic point; this is the versal variety defined above. Even once this is done, the versal torsor is not unique, nor are the rational maps mapping to it. We have the motto: “universal - unique = versal.” Versal varieties are useful for studying essential dimension. If X is a versal G-variety then ed(X) = ed(G) [7, Corollary 6.16]. If X  Y is a  compression of faithful G-varieties and X is versal then so is Y [7, Corollary 6.14]. Thus, if a versal variety exists, there exists a versal variety X such that dim(X) = ed(X) = ed(G). 8  Recall that a linear G-variety is a linear representation of G regarded as a G-variety. Any faithful linear G-variety is versal [27, Example 5.4]. Thus versal varieties exist. In particular, the essential dimension of any finite group is bounded above by the dimension of a faithful linear representation. The versal property descends to subgroups: Proposition 2.2. Suppose H is a subgroup of a finite group G. If X is a G-versal variety then X is H-versal. Proof. Clearly, a faithful G-action restricts to a faithful H-action. Consider any faithful H-variety Y and any non-empty H-invariant open subset U of X. We need to show the existence of an H-equivariant rational map f :Y  U . The set U = ∩g∈G g(U ) is a G-invariant dense open subset of U .  Since X is G-versal, there exists a G-equivariant rational map ψ : V  U  from a faithful linear G-variety V . Let W be a non-empty H-invariant open subset on which ψ is defined. Note that V is H-versal since the restricted action still acts linearly. Thus there exists an H-equivariant rational map φ:Y  W . By composition, we obtain an H-equivariant map f : Y  U  as desired. The following result is one of our major tools: Proposition 2.3. Let G be a finite group. If X is a proper versal G-variety then all abelian subgroups of G have fixed points on X. Proof. Recall the Going Down Theorem ([51, Proposition A.2]): Suppose A is a finite abelian group, Y  X is an A-equivariant rational map of  A-varieties where X is proper. If Y has a smooth A-fixed point then X has an A-fixed point. Now, if X is a proper versal G-variety, then there exists a rational Gequivariant map from a linear G-variety V . The origin is a smooth fixed point of any linear G-variety V . Thus, upon restriction to any abelian subgroup A, we have a fixed point on X by the Going Down Theorem. We recall various standard results on essential dimension which can be found in [12]. We say that a dihedral group, D2n , of order 2n is an odd dihedral group if n is odd, and an even dihedral group otherwise. 9  Proposition 2.4. Let G be a finite group. (a) If H is a subgroup of G then ed(H) ≤ ed(G). (b) If G is abelian then ed(G) = rank(G). (c) ed(G) = 0 if and only if G is trivial. (d) ed(G) = 1 if and only if G is cyclic or odd dihedral. The covariant dimension of a group G, denoted covdim(G), is the minimal dimension of a faithful G-variety X such that there is a faithful linear G-variety V and a dominant regular G-equivariant map V → X. One may consider covariant dimension as a regular analog of essential dimension. The interested reader is directed to the work of Kraft, L¨otscher and Schwarz ([40], [39]). The following result follows from the classification of groups of covariant dimension 2. We do not use the concept of covariant dimension anywhere else in this thesis. Proposition 2.5. If G is a finite group of essential dimension 2 with a nontrivial centre then G is isomorphic to a subgroup of GL2 (C). In particular, G has a versal action on P2 . Proof. By [39, Proposition 3.6], whenever G has a non-trivial centre we have ed(G) = covdim(G). By [39, Section 7], all finite groups of covariant dimension 2 are isomorphic to subgroups of GL2 (C). Thus we have a faithful linear G-variety of dimension 2. This is versal and equivariantly birational to P2 . We remark that, since all non-trivial p-groups have non-trivial centres, this proposition suffices to prove the Karpenko-Merkurjev theorem for groups of essential dimension 2. Recalling that all irreducible representations of p-groups have degree a power of p, we have the following: Proposition 2.6. If p > 2 is a prime, then all p-groups of essential dimension 2 are abelian.  10  2.2  Minimal Rational Surfaces  Recall that a variety X is rational if it is birationally equivalent to Pn for n = dim(X). From an algebraic standpoint, this means that the function field of X is isomorphic to k(x1 , . . . , xn ) where the xi ’s are algebraically independent indeterminates. A variety X is unirational if there exists a dominant rational map Pn  X for some integer n. Note that X rational  implies that X is unirational. In fact, the properties are equivalent for curves (L¨ uroth’s theorem), and for complex surfaces (Castelnuovo’s theorem). We recall some basic facts about minimal rational surfaces (see [25] or [46]). Throughout this thesis, a surface is an irreducible non-singular projective 2-dimensional variety over C. A minimal G-surface is a faithful G-surface X such that any birational regular G-map X → Y to another faithful G-surface Y is an isomorphism. There is a (not necessarily unique) minimal G-surface in every equivariant birational equivalence class of Gsurfaces. The possible minimal G-surfaces are classified as follows: Theorem 2.7 (Enriques, Manin, Iskovskikh). If X is a minimal rational G-surface then X admits a conic bundle structure or X is isomorphic to a del Pezzo surface. Our interest in minimal rational surfaces is justified by the following proposition (see [58, §3.6]): Proposition 2.8. Suppose G is a finite group. Then ed(G) ≤ 2 if and only if there exists a minimal rational versal G-surface X. We will use analogous ideas for three-dimensional varieties in Chapter 7.  2.3  Polyhedral Groups  The following facts will be used extensively in the sections that follow. Most of these results can be found in, for example, [20]. Recall that a polyhedral group is a finite subgroup of PGL2 (C). Equivalently, the polyhedral groups are precisely the finite groups acting regularly on P1 . The polyhedral groups 11  were classified by Klein as follows: Cn , the cyclic group of n elements; D2n , the dihedral group of order 2n; A4 , the alternating group on 4 letters; S4 , the symmetric group on 4 letters; and A5 , the alternating group on 5 letters. These groups have normal structures as follows: Proposition 2.9. Suppose P is a polyhedral group and N is a non-trivial proper normal subgroup of P . We have the following possibilities: (a) P  S4 , N  A4 , P/N  (b) P  S4 , N  C2 × C2 , P/N  S3 ,  (c) P  A4 , N  C2 × C2 , P/N  C3 ,  (d) P  D2n , N  Cm , P/N  D2n/m where m|n,  (e) P  D4n , N  D2n , P/N  C2 ,  (f ) P  Cn , N  Note that D2  Cm , P/N C2 and D4  C2 ,  Cm/n where m|n. C2 × C2 are included above as degenerate  cases. Finally, we will need the following fact about lifts of polyhedral groups to 2-dimensional representations: Proposition 2.10. A finite subgroup G of PGL2 (C) has an isomorphic lift in GL2 (C) if and only if G is cyclic or odd dihedral.  12  Chapter 3  Versal Actions on Toric Varieties In Chapter 4 we will use the theory of toric varieties extensively to prove Theorem 4.5. Many of the results we use are applicable beyond the case of surfaces so we consider the case of versal actions on toric varieties in general.  3.1  Cox Rings and Universal Torsors  We recall the theory of toric varieties from [26], and Cox rings from [19]. We will also use the language of universal torsors from [16]. Note that the similarity of the terms “universal torsor” and “versal torsor” is merely an unfortunate coincidence. Given a lattice N  Zn , a fan ∆ in N is a set of strongly convex rational  polyhedral cones in N ⊗ R such that every face of a cone in ∆ is in ∆ and the intersection of any two cones in ∆ is a face of each. Given a fan one may construct an associated toric variety. The associated toric variety X = X(∆) contains an n-dimensional torus T = N ⊗ C× . The variety X is non-singular if every cone in ∆ is generated by a subset of a basis for N . The variety X is complete if the support of the fan is all of N ⊗ R. In this chapter, we will restrict our attention to complete non-singular toric varieties. Let M = Hom(N, Z) be the dual of the lattice N . Let DivT (X) be the group of T -invariant divisors of X. Let ∆(1) be the set of rays in the fan ∆. To each ray ρ ∈ ∆(1) we may associate a unique prime T -invariant divisor Dρ . In fact, ∆(1) is a basis for DivT (X). We have the following  13  exact sequence: 1 → M → DivT (X) → Pic(X) → 1 where Pic(X) is the Picard group of X. Denote K = Hom(Pic(X), C× ) and apply Hom(·, C× ) to the above sequence to obtain another exact sequence: 1 → K → (C× )∆(1) → T → 1 . From [19], any toric variety X has an associated total coordinate ring, or Cox ring, which we denote Cox(X). The ring Cox(X) is a Pic(X)-graded polynomial ring Cox(X) = C[xρ : ρ ∈ ∆(1)] and has an induced K-action via the grading. (Note that Cox uses G where we write K). The variety V = Spec(Cox(X)) is isomorphic to affine space C∆(1) and there is a closed subset Z ⊂ V obtained from an “irrelevant ideal.” The open subset V − Z is invariant under the K-action and, since X is non-singular, the map (V − Z) → X is a K-torsor. Indeed, this torsor is a universal torsor over X. We define Aut(X) as the normaliser of K in the automorphism group of V − Z. From [19, Theorem 4.2], there is an exact sequence 1 → K → Aut(X) → Aut(X) → 1 where we denote the last map π : Aut(X) → Aut(X). Let Aut(N, ∆) denote the subgroup of GL(N ) which preserves the fan ∆ (permutes the cones). The group Aut(N, ∆) has an isomorphic lift to Aut(X) via permutations of the basis elements {xρ }. The group (C× )∆(1) is a subgroup of Aut(X) which descends to T ⊂ Aut(X). More generally, if G is a group with a faithful action on X then there is a group E = π −1 (G) ⊂ Aut(X)  14  acting faithfully on V − Z. We have an exact sequence of groups 1 → K → E → G → 1.  (3.1)  For finite groups G, the group E acts as a subgroup of GL(V ): Lemma 3.1. Let G be a finite group acting faithfully on X. Then E acts linearly on V . Proof. From [19, Section 4], the linear algebraic group Aut(X) is of the form (Ru  Gs ) · Aut(N, ∆) where Ru is unipotent and Gs is reductive. (Note  that Cox’s notation Gs has nothing to do with the group G in our context). Since G is finite and K consists of semisimple elements, all elements of E are semisimple. Thus E ⊂ Gs · Aut(N, ∆). The group Gs is of the form Gs =  GL(Sαi )  where the Sαi s are the weight-spaces of the action of K on V (as a vector space). The group Aut(N, ∆) permutes the basis vectors of V . Thus Gs · Aut(N, ∆) acts linearly on V . Thus the subgroup E acts linearly. The versal property is related to Cox rings by the following result: Theorem 3.2. Suppose G is a finite group and X is a complete non-singular toric faithful G-variety. Then X is versal if and only if the exact sequence (3.1) splits. Proof. Suppose the exact sequence splits. The map from V − Z to X may be viewed as a dominant rational map ψ : V  X. Since E is linear for  any finite group G, we obtain an E-equivariant rational map from a linear E-variety to X. Since there is a section G → E the map ψ may be viewed as a G-equivariant dominant rational map from a linear G-variety. Thus X is versal. For the other implication, we assume X is versal and want to show (3.1) splits. Since X is versal there exists a G-equivariant rational map f : W  X  where W is a faithful linear G-variety. Let P → U be the K-torsor obtained 15  by pulling back ψ along the restriction of f to its domain of definition U . From the universal property of pullbacks we obtain an E-action on P compatible with the G-action on U . Note that Pic(U ) = 0 since U is open in the affine space W . Thus, from the exact sequence [16, (2.0.2)], we see that the ´etale cohomology group H 1 (U, K) is trivial. In particular, the torsor P → U is trivial. Since X is proper and U is normal, the indeterminacy locus of f is of codimension ≥ 2. Thus, all invertible global functions on U are constant and the space of sections of P → U is isomorphic to K. Thus E has an induced action on K and the desired splitting follows from Lemma 3.3 below. Lemma 3.3. Suppose E is an algebraic group with closed normal subgroup K and quotient G = E/K. Suppose E acts on K such that the restricted action of K on itself is translation. Then E splits as K  G.  Proof. Take any point p ∈ K and consider the stabiliser S = StabE (p). For any g ∈ G we have a lift h ∈ E. There is an element k ∈ K such that kh(p) = p. Thus kh ∈ S and it follows that S/(S ∩ K) = G. Since K acts freely on itself, S ∩ K = 1. Thus S  G and we have a splitting of E.  Remark 3.1. While Cox rings were originally introduced only for toric varieties, they can be defined in more general contexts (see, for example, [30] and [41]). Indeed, the notion of universal torsors were introduced with other varieties in mind [16]. The Cox rings and universal torsors of minimal rational surfaces have been extensively studied. For example, conic bundles are considered in [16, §2.6]; del Pezzo surfaces, in [23] and [56]. It would be interesting to investigate versality using these constructions. In fact, the proof of Theorem 3.2 still applies in one direction: if a Gaction on a complete non-singular variety is versal then an analogous exact sequence to (3.1) would still split. However, the analog of V is linear if and only if X is toric [30, Corollary 2.10]. Thus, in general, one does not have an obvious compression from a linear G-variety as above. Recall that the standard projection Cn  Pn−1 is an example of ψ  obtained from the Cox ring. We point out a special case of the preceding 16  proposition: Corollary 3.4. Let G be a finite group acting faithfully on X = Pn−1 . Then X is G-versal if and only if there exists an embedding G → GLn (C) such that the composition with the canonical map GLn (C) → PGLn (C) gives the G-action on Pn−1 . Remark 3.2. Theorem 3.2 and Corollary 3.4 were inspired by Ledet’s classification of finite groups of essential dimension 1 over an infinite ground field k [42]. Indeed, [42, Theorem 1] states that a finite group G has essential dimension 1 if and only if there is an embedding G → GL2 (k) such that the image of G contains no scalar matrices = 1. Such a group descends isomorphically to a subgroup of PGL2 (k). In other words, the action of G on P1k lifts to A2k . The following is a useful tool for showing that a variety is versal. Corollary 3.5. Suppose G is a finite group acting faithfully on a complete non-singular toric variety X. If G has a fixed point then X is G-versal. Proof. We have an action of E on the fibre of the fixed point, so the result follows from Lemma 3.3. Remark 3.3. We note that Corollary 3.5 fails when X is not toric. Consider a hyperelliptic curve C with its involution (generating a group G  C2 ).  This is a faithful G-variety with a fixed point. However, C cannot be versal since the image of a rational map from a linear variety to C must be a point. The following corollary is also inspired by Ledet [42]: Corollary 3.6. Let G be a finite group acting faithfully on a complete nonsingular toric variety X. Then X is G-versal if and only if, for any prime p, X is Gp -versal for a Sylow p-subgroup Gp of G. Proof. Using Theorem 3.2 this follows from a well-known result in group cohomology. Consider the product  G p ResGp  of the restriction maps ResG Gp :  H 2 (G, K) → H 2 (Gp , K) over all primes p and some choice of Sylow psubgroups Gp for each p. From [11, Section III.10], this product is an injection. Thus E → G has a section if and only if every Gp has a section. 17  We remark on one application of this corollary that is not immediately obvious, but extremely useful. Suppose X is a G-variety and we want to determine whether or not it is versal. For each prime p, let Gp be a p-Sylow subgroup of G. Suppose X is Gp -equivariantly birational to a Gp -variety Xp for each prime p. The versality property may be easier to determine on the new varieties Xp than on the original variety X. This corollary is our main tool in the proof of Theorem 4.5. In particular, we will show that the versality question on all toric surfaces can be reduced to studying 3-groups acting on P2 and 2-groups acting on P1 × P1 .  3.2  Monomial Actions  We make the following observation: Lemma 3.7. Suppose X is a toric variety with a faithful action of a finite group G contained in the torus T . Then X is G-versal. Furthermore, if X is complete, then X has a G-fixed point. Proof. First, suppose X is complete; by the Borel fixed point theorem X has a T -fixed point and, thus, a G-fixed point. In general, X is T -equivariantly birationally equivalent to a complete non-singular toric variety (say Pn ). Consequently, this birational equivalence is G-equivariant. Thus X is Gversal by Corollary 3.5. Consider a toric variety X with torus T , fan ∆ and lattice N . Recall that Aut(N, ∆) is the subgroup of GL(N ) preserving the fan ∆. Note that the group Aut(N, ∆) has a natural action on X which is T -stable. We say a group G has a multiplicative action on X if G ⊂ Aut(N, ∆). Lemma 3.8. Suppose X is a toric variety with a faithful multiplicative action of a finite group G. Then X has a G-fixed point and X is G-versal. Proof. Any element of Aut(N, ∆) fixes the identity of the torus T in X. The versality of X is well-known (see [15, Lemma 3.3(d)] or [2]).  18  Note that both T -actions and multiplicative actions are T -stable — they preserve T as a subvariety of X. Any particular T -stable automorphism of X is a product of an element of T and an element of Aut(N, ∆). Thus, the group of T -stable automorphisms of X is precisely AutT (X) = T  Aut(N, ∆) .  Given such a subgroup of AutT (X) there is a natural map ωT : G → Aut(N, ∆) ⊂ GL(N ) given by the projection G → G/(G ∩ T ). We denote this map ωT to emphasize its dependence on T (even though, strictly speaking, it depends on N ). Despite the fact that T -actions and multiplicative actions are always versal, this does not hold for T -stable actions in general. Nevertheless, they are much more manageable than general actions. Definition 3.1. Let G be a group acting faithfully on a toric variety X. We say that the action is monomial if there exists a fan ∆ in a lattice N inducing a torus T = N ⊗ C× such that the associated toric variety is G-equivariantly biregular to X and g(T ) = T for all g ∈ G. Such actions are also called “twisted multiplicative” in the literature. Note that, for a linear variety X  Cn , monomial actions are precisely  the same as monomial representations. Recall that all linear representations of supersolvable groups are monomial [57, Section 8.5, Theorem 16]. This result has a natural generalisation for toric varieties. Proposition 3.9. Suppose G is a supersolvable finite group acting on a complete non-singular toric variety X. Then G is monomial. Proof. By Lemma 3.10 below, there exists a change of basis α : V → V such that E = π −1 (G) has a monomial action on α(V ) with K acting diagonally. Since K acts diagonally in both coordinates, if Vλ is the weight space in V  19  corresponding to some character λ : K → C× then α(Vλ ) ⊂ Vλ . These Vλ are precisely the Sαi of [19, Section 4]. This means that α ∈ Gs where Gs =  GL(Sαi ) ⊂ Aut(X).  Thus α descends to an automorphism of X. In the new basis, we have an embedding E → (C× )∆(1) Aut(N, ∆). Taking the quotient by K we obtain G⊂T  Aut(N, ∆).  Lemma 3.10. Suppose we have an exact sequence of algebraic groups (over C) 1→K→E→G→1 where K is diagonalisable and G is finite supersolvable. For any representation V of E there exists a choice of coordinates such that E is monomial with K diagonal. Furthermore, any irreducible representation has dimension dividing the order of G. Proof. This is a straight-forward generalisation of [57, Section 8.5, Theorem 16]. We proceed by induction on the dimension of V . For any normal subgroup N  E the quotient η : E → E/N sits in an exact sequence 1 → η(K) → η(E) → η(E)/η(K) → 1  with η(K) diagonalisable and η(E)/η(K) finite supersolvable. Thus it suffices to assume V is a faithful irreducible representation of E. Suppose E is abelian. There are no non-trivial unipotent elements in G or K, so E consists of semisimple elements. Thus E is diagonalisable (thus monomial). This also takes care of the base case dim(V ) = 1. Suppose E is non-abelian. We claim there exists a normal diagonalisable subgroup A containing K which is not contained in the centre of E. If K is not central we may take A = K. If K is central then there exists a normal cyclic subgroup C of E/Z(E) by supersolvability of E/K. In this case, take A to be the inverse image of C in E. We see that A is abelian (thus diagonalisable since K is diagonalisable), contains K, and is not contained 20  in the centre of E. We have a decomposition V = ⊕Vi into distinct weight spaces for the action of A. Since A is normal in E, the group E permutes the spaces Vi . In fact, the action of E is transitive since V is irreducible. Since E acts faithfully on V and A is not central in E, there is more than one weight space Vi . Let H be the maximal subgroup of E such that H(V0 ) = V0 . We see that the E-representation V is induced from the H-representation V0 . Since dim(V ) = [E : H] dim(W ) and H contains K the result follows from the induction hypothesis. Recall that p-groups are supersolvable. Thus, in particular, actions of p-groups on toric varieties are always monomial. This is particularly useful in light of Corollary 3.6 above.  21  Chapter 4  Del Pezzo Surfaces of Degree ≥5 The main goal of this section is to prove Theorem 4.5: a classification of all finite groups which act versally on one of the four surfaces P2 , P1 × P1 , DP6 (the del Pezzo surface of degree 6) or DP5 (the del Pezzo surface of degree 5). Recall that the automorphism group of P2 is PGL3 (C); that of P1 × P1 is (PGL2 (C) × PGL2 (C))  S2  where S2 swaps the two copies of PGL2 (C); that of DP6 is (C× )2  D12 (see  [25, Section 6.2]); and that of DP5 is S5 (see [25, Section 6.3]). The surfaces P2 , P1 × P1 and DP6 are toric. The monomial actions on these surfaces will be particularly important. For example, we have the following lemma: Lemma 4.1. All versal actions of finite groups on P1 × P1 are monomial. Proof. Recall that π : Aut(X) → Aut(X) is the group homomorphism induced from the Cox ring construction. For G ⊂ Aut(X) we have the lift E = π −1 (G) in Aut(X)  (GL2 (C) × GL2 (C))  S2  with the exact sequence (3.1) from Chapter 3. Let H = G ∩ (PGL2 (C) × PGL2 (C)). The group H is the image in G of the centralizer of K in E. We see that H is a normal subgroup of G of index at most 2. Let H1 and H2 be the projections of H to the first and second copies of PGL2 (C). We note that there is a natural embedding 22  H ⊂ H1 × H2 . When H = G we have isomorphisms H1  H2 induced by  the actions of elements in G − H. We may consider the action of E on V  C4 as a 4-dimensional repre-  sentation ρ. Let EH = π −1 (H). Note that EH ⊂ GL2 (C) × GL2 (C). Thus, the restriction ρ|EH is a direct sum of 2-dimensional subrepresentions σ1 and σ2 . Informally, one may consider σ1 as the preimage of H1 and σ2 as the preimage of H2 . If G = H then ρ is induced from σ1 . If G is versal then there is a section from G to E. Recall that a finite subgroup of PGL2 (C) lifts isomorphically to GL2 (C) if and only if it is cyclic or odd dihedral (Proposition 2.10). All 2-dimensional representations of lifts of such groups are monomial. Thus σ1 and σ2 are monomial. If G = H then ρ = σ1 ⊕ σ2 is monomial. If G = H then ρ = IndG H σ1 is monomial.  4.1  Monomial Actions on Toric Surfaces  Recall the classification of conjugacy classes of finite subgroups of GL2 (Z). We use the list in Lorenz’s book [43, page 30]. The list comes with explicit representatives for each conjugacy class in terms of explicit matrix generators. We use the Gi notation to denote this explicit representative in each conjugacy class. (Lorenz uses Gi to denote the class, not the representative). Since it is used so extensively in what follows, we reproduce the list in Table 4.1. One checks that Figure 4.1 contains the finite subgroup lattice structure in GL2 (Z) where an arrow means “contains a subgroup in the conjugacy class of”. We omit composite arrows for clarity. From this subgroup lattice structure we make some useful observations about p-groups in GL2 (Z). For p > 3, there are no non-trivial p-subgroups of GL2 (Z). All 2-subgroups of GL2 (Z) are conjugate to a subgroup of G2 . All non-trivial 3-subgroups of GL2 (Z) are conjugate to G9 . Let N = Z2 be our lattice. There are standard realisations of P2 , P1 × P1 and DP6 as the toric varieties associated to the complete fans in N from Figure 4.2. Let T = N ⊗ C× be the torus associated to the lattice N . Choose 23  Label  Generators  Structure  G1  1 −1 , 0 1 10 1 0 −1 0 , 0 1 10 0 1 0 −1 0 −1 , 1 −1 −1 0 0 −1 01 1 −1 , 1 0 −1 0 , 1 0 0 −1 0 1 0 1 , 0 −1 10 −1 0 1 −1 1 0 0 −1 1 0 0 −1 1 −1 −1 0 0 −1 −1 0 0 1 01 10  D12  G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12  D8 D6 D6 C2 × C2 C2 × C2 C6 C4 C3 C2 C2 C2  Table 4.1: Conjugacy classes of non-trivial finite subgroups of GL2 (Z)  G1  G3  G2  ❍ ❅ ❍❍     ❅ ❍❍ ❍❍ ✠   ❘ ❅ ❄ ❥  G4  ❍ ❅ ❍❍ ❅   ❅ ❍❍   ❅ ❘ ❄  ❅ ✠❍❍ ❘ ❅ ❥  G9  G7      ✠    G6  ❅ ❅ ❘ ❅  ❄  G8  G5 ✟✟  G12  ❅     ✟   ❅   ✟✟ ✠ ❅   ❘ ❄  ✠ ✟✟ ✙  G10  ❄  G11  Figure 4.1: Lattice of finite subgroups in GL2 (Z)  24  (0 : 0 : 1)  (1 : 0; 0 : 1)  (0 : 1; 0 : 1)  (1 : 0; 1 : 0)  (0 : 1; 1 : 0)  (1 : 0 : 0)  (0 : 1 : 0)  ∆P2  ∆P1 ×P1  ∆DP6  Figure 4.2: Standard fans for P2 , P1 × P1 and DP6  25  coordinates such that (λ1 , λ2 ) ∈ (C× )2  T corresponds to  (λ1 : λ2 : 1) ∈ P2 and (λ1 : 1 ; λ2 : 1) ∈ P1 × P1 . Thus the maximal cones in ∆P2 and ∆P1 ×P1 correspond to the T -fixed points indicated in the diagram. Recall that the group of T -stable automorphisms AutT (X) of a surface X is T  Aut(N, ∆X ) where Aut(N, ∆X ) is the group of automorphisms of  the associated fan ∆X . We have the following automorphism groups: Aut(N, ∆P2 ) = G4  Aut(N, ∆P1 ×P1 ) = G2  Aut(N, ∆DP6 ) = G1  Since G1 and G2 are the maximal finite subgroups of GL2 (Z) up to conjugacy (see Figure 4.1), all monomial group actions on toric surfaces are equivariantly birational to actions on P1 × P1 or DP6 . (Note, however, that this is not the case for general actions.) By Lemma 3.7, this means that all faithful p-group actions on toric surfaces are automatically versal for p > 5. For 3-groups and 2-groups the theory is a bit more involved. For 3-groups, the versal property can be determined by considering actions on P2 : Lemma 4.2 (3-groups acting on toric surfaces). Suppose G3 is a finite 3group acting faithfully on a toric surface X. Then X is G3 -birationally equivalent to Y = P2 with G3 ⊂ T  G9 and the following are equivalent:  1. Y has a G3 -fixed point, 2. X is G3 -versal, 3. the following conditions hold: (a) if ωT (G3 ) = 1 then there are no conditions, (b) if ωT (G3 ) = G9 then G3 ∩ T = 1.  26  Proof. Recall that all p-groups are monomial and G9 is the maximal finite 3-subgroup of GL2 (Z) up to conjugacy. Thus we may assume G3 ⊂ T  G9  in some coordinates by Proposition 3.9. Furthermore, selecting a new fan in the same lattice induces a birational map. Thus X is G3 -birationally equivalent to Y = P2 . Since versality is a G-birational invariant, it suffices to assume X = Y for the remainder of the proof. The implication (1) =⇒ (2) is immediate by Corollary 3.5. In case (a), all remaining implications are immediate by Lemma 3.7. It remains to consider case (b) with ωT (G3 ) = G9 . Assume (2): that X is versal. Since G3 is a 3-group of essential dimension 2, it is abelian by Proposition 2.6. Note that G9 does not fix any of the cones of the fan ∆P2 except for the trivial cone {0}. Thus any G3 -fixed point must be on the torus. Note the action of any non-trivial element of G3 ∩ T does not fix any point on the torus. If G3 ∩ T = 1 then we have an abelian subgroup without a fixed point. This contradicts Proposition 2.3. So (3) must hold and we have (2) =⇒ (3). If (3) holds, then G3  C3 . Any finite cyclic group acting on P2 has a  diagonalisation. So there exists a G3 -fixed point. We have (3) =⇒ (1). Similarly, we determine which 2-groups are versal by studying P1 × P1 . Lemma 4.3 (2-groups acting on toric surfaces). Suppose G2 is a finite 2-group acting faithfully on a toric surface X. Then X is G2 -birationally equivalent to Y = P1 × P1 with G2 ⊂ T  G2 and the following are equivalent:  1. Y has a G2 -fixed point, 2. X is G2 -versal, 3. the following conditions hold: (a) if ωT (G2 ) is conjugate to 1 or G12 then there are no conditions, (b) if ωT (G2 ) is conjugate to G11 then (after choosing coordinates such that ωT (G2 ) = G11 ), for any t ∈ G2 ∩ T , t = (1, λ) for some λ ∈ C× , 27  (c) in all remaining cases we require G2 ∩ T = 1. Proof. Recall that G2 is the maximal finite 2-subgroup of GL2 (Z) up to conjugacy. Similarly to Lemma 4.2 above, we may assume G2 ⊂ T  G2 and  X =Y. The implication (1) =⇒ (2) is immediate by Corollary 3.5. We prove the remaining implications by restricting to each case in (3). Case (a): ωT (G2 ) is conjugate to 1 or G12 . There are no additional conditions so it suffices to show (1) always holds. When ωT (G2 ) = 1, this is immediate from Lemma 3.7. For ωT (G2 ) conjugate to G12 we choose coordinates so that ωT (G2 ) = G12 and use the fan ∆P1 ×P1 as above. The cone σ spanned by {(1, 0), (0, 1)} is fixed by the action of G12 and, since it is a maximal cone, corresponds to a T -fixed point. Thus the T -orbit corresponding to σ is a G-fixed point. Case (b): ωT (G2 ) is conjugate to G11 . It suffices to assume that ωT (G2 ) = G11 and Y is constructed from the fan ∆P1 ×P1 . Assume (3) does not hold; we will show that this implies (2) cannot hold. There exists an element t ∈ G2 ∩ T of the form t = (λ1 , λ2 ) ⊂ (C× )2 where λ1 = 1. Furthermore, by taking appropriate powers, we may assume λ1 has order 2. Now consider g ∈ G2 such that ωT (g) = 1. The group A = g, t is an abelian subgroup of G2 . Note that G11 (and thus g) only fixes the cones spanned by {(0, 1)}, {(0, −1)} and {0} in ∆P1 ×P1 . The element t acts non-trivially on the T -orbits corresponding to those cones (and so has no fixed points there). We have an abelian subgroup A without a fixed point. This contradicts Proposition 2.3. Thus (2) does not hold. We have shown (2) =⇒ (3). Now assume (3) holds. Recall the definitions of H, H1 and H2 from the proof of Lemma 4.1. In this case G2 = H; and H1 , H2 are cyclic. Thus H1 has a fixed point p1 on the first P1 and and H2 has a fixed point p2 on the second. The point (p1 , p2 ) is a G2 -fixed point of Y . Thus (3) =⇒ (1) =⇒ (2). Case (c): all remaining cases. Assume (3) does not hold. Recall the subgroup structure of G2 from 28  figure 4.1. We must have G10 ⊂ ωT (G2 ). If G2 ∩ T = 1 then there exists an element t ∈ G2 ∩ T of order 2. The element t commutes with the action of G10 . Let g ∈ G2 be an element such that 1 = ωT (g) ⊂ G10 . The group A = g, t is an abelian subgroup of G2 . Note that G10 only fixes the trivial cone {0} in ∆P1 ×P1 so any A-fixed point must be on the torus. The element t does not fix any point on the torus. We have an abelian subgroup A without a fixed point. This contradicts Proposition 2.3. Thus (2) does not hold. We have shown (2) =⇒ (3). Now assume (3) holds. In this case, H1 and H2 are cyclic. A cyclic subgroup of PGL2 (C) lies in some torus C× . Thus we may find new coordinates with a different torus T ⊂ Y such that H ⊂ T . Note that, for any g ∈ G2 , g 2 ∈ T so ωT (g) has order 2. Also, G10 and G11 are contained in H. Thus, any g ∈ G2 − H must have ωT (g) conjugate to the non-trivial element in G12 . So G2 ⊂ T  G12 and has a fixed point by case (a). We have shown  (3) =⇒ (1) =⇒ (2). Lemma 4.4. All finite subgroups G of the following groups have versal monomial actions on a toric surface: (1*) T  G12 .  (2) T  G1 with |G ∩ T | coprime to 2 and 3,  (3) T  G2 with |G ∩ T | coprime to 2,  (4) T  G3 with |G ∩ T | coprime to 3,  (5) T  G4 with |G ∩ T | coprime to 3,  Furthermore, any finite group with a versal monomial action on a toric surface is of this form. Proof. Recall that when deciding whether a group G has a versal action on a complete non-singular toric variety it suffices to check Sylow p-subgroups Gp (Corollary 3.6). For all the forms above, Gp is always versal when p ≥ 5 by Lemma 3.7. So one only needs to check the Sylow 3- and 2-subgroups.  29  Any finite G with a monomial action can be written in the form G ⊂ T  Gi where Gi is from Table 4.1. From Lemma 4.2 and Lemma 4.3 we have  necessary and sufficient conditions for G2 and G3 to be versal. We note G3 ∩ T = 1 is equivalent to |G ∩ T | coprime to 3 and similarly for G2 . By selecting appropriate Sylow subgroups we have Table 4.2 where the last row gives the necessary and sufficient conditions for G to be versal. One sees that any group G listed in the theorem is versal. For the converse, we show that all of the other possibilities for ωT (G) are already contained in a group appearing in the list. The cases G5 , G6 , G8 and G10 are all covered by form (3); G7 , by form (2); G9 , by forms (4) and (5); and ωT (G) = 1 by form (1*). It remains to eliminate the special case G11 . Here, G = H ⊂ H1 × H2 in the language of the proof of Lemma 4.1. Thus any finite subgroup G of T  G11 must be a subgroup of D2n × Cm for sufficiently large integers n and  m. From case (3b) of Lemma 4.3, if G is versal we can assume n is odd. We show that any such group is actually isomorphic to a group of form (1*) above. Indeed, consider the following subgroup of GL2 (C): ζn 0 −1 0 ζn  ,  ζm 0 0 ζm  ,  01 10  where ζn and ζm are nth and mth roots of unity, respectively. This group is isomorphic to D2n × Cm and has an embedding into T  4.2  G12 .  Versal Actions on the Four Surfaces  Theorem 4.5. Suppose a finite group G has a versal action on P2 , P1 × P1 , DP6 , or DP5 . Then G is finite subgroup of one of the following groups: (1) GL2 (C), (2) T  G1 with |G ∩ T | coprime to 2 and 3,  (3) T  G2 with |G ∩ T | coprime to 2,  30  ωT (G)  ωT (G3 )  ωT (G2 )  |G ∩ T | coprime to  G1  G9  G6  2, 3  G2  1  G2  2  G3  G9  G12  3  G4  G9  G12  3  G5  1  G5  2  G6  1  G6  2  G7  G9  G10  2, 3  G8  1  G8  2  G9  G9  1  3  G10  1  G10  2  G11  1  G11  special  G12  1  G12  none  1  1  1  none  Table 4.2: Versality conditions for monomial actions on surfaces  31  (4) T  G3 with |G ∩ T | coprime to 3,  (5) T  G4 with |G ∩ T | coprime to 3,  (6) PSL2 (F7 ), (7) S5 . Furthermore, all finite subgroups of the above groups act versally on one of those surfaces. Proof. Recall that any finite subgroup of GL2 (C) acts versally on P2 . We note that any finite subgroup of T  G12 is a subgroup of GL2 (C). So form  (1*) in Lemma 4.4 is wholly contained in form (1) of this theorem. By Lemma 4.4, the versal monomial actions on any toric surface are contained in forms (1)–(5) above. Recall that the automorphism group of DP6 is T T  G1 and the group of monomial automorphisms of P1 × P1 is  G2 . Forms (1)–(5) all have versal actions on one of the 4 surfaces. It remains to study actions that are not monomial. All actions on DP6  are monomial, and by Lemma 4.1, this is also true of versal actions on P1 × P1 . Thus these surfaces require no more consideration. We consider DP5 . Recall from [25, Section 6.2] that a del Pezzo surface of degree 5 can be described as a quotient (P1 )5 / PSL2 (C). The automorphism group of DP5 is S5 and its action is versal by the construction in [12]. Thus, all subgroups of S5 act versally on DP5 . It remains to classify all finite groups acting versally on P2 . Recall that, by Corollary 3.4, it suffices to determine whether there is an isomorphic lift from PGL3 (C) to GL3 (C). We refer to Blichfeldt’s classification of finite subgroups of GL3 (C) in [9, Chapter V]. Using Blichfeldt’s notation, we note that groups of types A and B descend to subgroups of GL2 (C), and groups of type C and D descend to monomial actions on P2 . These groups have already been considered. Finally, we consider the exceptions E–J in the classification. Blichfeldt appends the symbol “φ” to the order of a subgroup of GL3 (C) when there is no isomorphic lift of its image in PGL3 (C). Consequently, only types H 32  and J descend to versal actions — these correspond to the groups A5 and PSL2 (F7 ).  33  Chapter 5  Conic Bundle Structures Recall Manin and Iskovskikh’s classification of minimal rational G-surfaces into conic bundles and del Pezzo surfaces from Section 2.2. In this section, we establish the conic bundles case of Theorem 1.2. The del Pezzo surfaces case will be considered in Chapter 6. Theorem 5.1. If G has a versal action on a minimal conic bundle X then G has a versal action on P1 × P1 or P2 . All of the following facts about conic bundle structures can be found in [25, Section 5] or [33]. A conic bundle structure on a rational G-surface X is a G-equivariant morphism φ : X → B such that B  P1 and the fibres  are isomorphic to reduced conics in P2 . Note that, unlike del Pezzo surfaces, the G-action is required for this definition to make sense. There may exist other group actions where X does not have such a structure (for example, not all actions on P1 × P1 respect the fibration). A fibre F of the morphism φ is either isomorphic to P1 or to P1 ∧ P1 (two copies of P1 meeting at a point). In the first case, Aut(F )  PGL2 (C); in  the second, Aut(F ) has a monomial representation of degree 2 (in particular, it is a subgroup of GL2 (C)). Let Fn be the ruled surface P(OP1 ⊕ OP1 (n)) for a non-negative integer n (see, for example, [28, Section V.2]). A conic bundle is either isomorphic to some Fn or to a surface obtained from some Fn by blowing up a finite set of points, no two lying in a fibre of a ruling. Let π : G → PGL2 (C) be the map induced by the action of G on B under φ. Let GB = im(π) and GK = ker(π). One may consider GK as the largest subgroup of G which preserves the generic fibre. Note that every fibre of φ is GK -invariant. It is useful to think of GK as the group that 34  “acts on the fibre” and GB as the group that “acts on the base.” Both GK and GB are polyhedral groups since they act faithfully on rational curves. Let Σ = {p1 , . . . , pr } be the set of points on B whose fibres are singular. Let R be the set of components of singular fibres {R1 , R1 , . . . , Rr , Rr } where Ri and Ri are the two components of the fibre φ−1 (pi ) for each pi ∈ Σ. We have a natural map ξ : G → Aut(R) where Aut(R) is the group of permutations of R. Let us denote G0 = ker(ξ) ∩ GK (note that this differs slightly from the definition in [25, Section 5.4]). Proof of Theorem 5.1. We prove the theorem by considering the different possibilities for GK . We suggest reviewing the results of Section 2.3. Note that if GK contains a characteristic subgroup of order 2 then G has a non-trivial centre and, thus, a versal action on P2 by Proposition 2.5. The polyhedral groups with characteristic subgroups of order 2 are the dihedral groups D4n with n ≥ 2, and the cyclic groups of even order. It remains to consider GK of the following types: odd cyclic, odd dihedral, C2 × C2 , A4 , S4 and A5 . By Lemma 5.3 below, G0 acts faithfully on every component of every fibre of φ. If S has no singular fibres then X is a ruled surface and we may apply Lemma 5.2. Consequently, we may assume π has a singular fibre F . So G0 acts faithfully on an irreducible component of F with a fixed point. Any such component is isomorphic to P1 . The only polyhedral groups with fixed points are the cyclic groups, so G0 is odd cyclic. Note that GK can only permute components of the same fibre, thus ξ(GK ) ⊂ (C2 )r . We have a normal structure with G0 GK /G0 ⊂ (C2  )r .  This excludes GK  A4 , GK  GK cyclic and  S4 and GK  A5 . Thus  it remains only to consider groups GK that are odd cyclic, odd dihedral or isomorphic to C2 × C2 . These remaining cases are handled by the lemmas below. If GK is odd cyclic then the result follows by Lemma 5.4 below; this case corresponds to X being a ruled surface. If GK is odd dihedral then Lemma 5.5 applies; these surfaces are the “exceptional conic bundles” of [25, Section 5.2]. Finally, if GK  C2 × C2 then Lemma 5.6 applies; these are all “non-exceptional conic 35  bundles” as in [25, Section 5.4]. Lemma 5.2. If X is a ruled surface with a versal G-action then G acts versally on P1 × P1 or P2 . Proof. If X is P1 × P1 then we are done. Otherwise, from [25, Theorem 4.10] we see that any finite group acting on a ruled surface is a central extension of a finite subgroup of PGL2 (C) or SL2 (C). Any finite subgroup G of such an extension has a versal action on P2 . Indeed, it suffices to consider G with trivial centre. Any such G then embeds into PGL2 (C) or SL2 (C). All polyhedral groups have versal actions on P2 (see proof of Theorem 4.5); as do all finite subgroups of SL2 (C). Lemma 5.3. The group G0 acts faithfully on every component of every fibre of φ. Proof. Let R be a component of a fibre of φ. Since G0 preserves components of fibres, we may G0 -equivariantly blowdown X to a ruled surface such that R is isomorphic to a fibre of the blowdown variety. Thus, it suffices to prove the theorem for X when all fibres are isomorphic to P1 . Let g be any non-trivial element of G0 . There exists an open cover of B by open sets U such that φ−1 (U )  U × P1 . Let V be the subset of distinct  triples of points in (P1 )3 . There is an isomorphism V → PGL2 (C) by taking the automorphism determined by the images of the three points 0, 1 and ∞. By composing this isomorphism with the restrictions g|U × {0}, g|U × {1} and g|U × {∞}, we obtain a map γg,U : U → PGL2 (C) which takes each point to the action of g on the fibre of φ. Let α : PGL2 (C) → C be the map defined by α:A→  Tr(A )2 det(A )  where A is any lift of A to GL2 (C). One easily checks that α is well-defined and is invariant on conjugacy classes. Furthermore, for any A ∈ PGL2 (C) of finite order, α(A) = 4 if and only if A = 1 (by diagonalisation).  36  The isomorphism φ−1 (U )  U × P1 is only determined up to conjugacy  in PGL2 (C). Gluing together each γg,U after composing with α we obtain a map γg : B → C. Since C is affine and B  P1 , the image of γg is a point.  Since G0 acts faithfully on X, there must be at least one fibre on which g acts non-trivially. Thus γg = 4 and g acts non-trivially on every fibre. Thus G0 acts faithfully on every fibre. Lemma 5.4. Suppose G acts versally on X and GK is odd cyclic. Then G acts versally on P1 × P1 or P2 . Proof. It suffices to consider G-minimal X. When X is a ruled surface then the result follows from Lemma 5.2. As we shall see, this is the only case that occurs. Suppose X is not a ruled surface. We will show that GK must contain an involution, contradicting the assumption that GK has odd order. We use the same reasoning as the proof of [25, Lemma 5.6]. Since X is G-minimal, there must exist an element g ∈ G that swaps two components, R and R , of a singular fibre of φ. By taking an odd power, we may assume that g has order m = 2a . Consider a = 1. The intersection point p of R and R is in the fixed locus X g . Any involution acting on a surface with an isolated fixed point must act via (x, y) → (−x, −y) in some local coordinates about that point. Thus, if p is an isolated fixed point then g cannot swap R and R . This contradiction insures that X g contains a curve other than the fibres of φ. Thus, g is contained in GK . Now, consider the remaining case a > 1. Consider h = g m/2 . Suppose X h contains R. Then hg(y) = gh(y) = g(y) applies for all y ∈ R. This means that R is contained in X h as well, contradicting the smoothness of X h . Thus, neither component is contained in X h . There exists exactly one fixed point y on R other than its intersection with R . If y was an isolated h-fixed point on X then its image q would still be an isolated h-fixed point upon blowing down R. But then R has a trivial h-action: a contradiction. Thus h fixes a curve not contained in the fibres of φ. We obtain h ∈ GK . 37  Lemma 5.5. Suppose G acts versally on X and GK is odd dihedral. Then G acts versally on P1 × P1 . Proof. Recall that G0  Cn and GK  D2n for some n odd as in the proof  of Theorem 5.1. Consider any g ∈ GB , we shall prove that H = π −1 ( g ) is a direct product GK × g . Since g is cyclic there is a fixed point on B. Thus φ has an H-fixed fibre F . Recall that G0 acts faithfully on F . If F is non-singular then PGL2 (C). If F is singular then Aut(F ) ⊂ GL2 (C) and we have a  Aut(F )  natural map Aut(F ) → PGL2 (C) with central kernel. The group G0 is not in the centre of GK , so we have map η : H → PGL2 (C) which is injective on G0 . Since η is injective on G0 it must be injective on GK . The image of η must be a polyhedral group with a normal subgroup isomorphic to GK  D2n for  some odd n. From Lemma 2.9, the only possibilities are η(H)  GK or  η(H)  GK × C2 (since n is odd). Either way, there exists a retract  D4n  H → GK of the inclusion GK → H. Thus H  GK × g .  Recall that GK has a trivial centre. By [11, Corollary IV.6.8], there is only one extension of GK by GB associated to a map GB → Out(GK ) (up to equivalence). Since g has a trivial action on GK for any g ∈ GB , the map GB → Out(GK ) is trivial. Thus we must have G  GK × GB .  The group GK contains an involution. If GB contains a subgroup isomorphic to C2 ×C2 then G contains (C2 )3 . This would contradict ed(G) ≤ 2. So GB must be cyclic or odd dihedral. Thus, G action on  P1  ×  P1  where GK acts on one  P1 ,  GK × GB has a versal  and GB , the other.  Lemma 5.6. Suppose G acts versally on X and GK acts versally on  C2 × C2 . Then G  P2 .  Proof. It suffices to consider G with a trivial centre, since otherwise we immediately have a versal action on P2 by Proposition 2.5. We have a map GB → Aut(GK )  S3 with kernel J. We note that, by construction, if g ∈ G  maps to J ⊂ G/GK then g commutes with GK . Suppose J = 1. If G → S3 is not surjective then G is abelian or isomorphic to A4 . Both of these have versal actions on P2 so it suffices to assume 38  G is an extension of C22 by S3 . A 2-Sylow subgroup of G is not normal, since we would obtain a non-trivial map from S3 to C3 (which cannot exist). A 3-Sylow subgroup of G is not normal since A4 ⊂ G and C3 is not normal in A4 . The only group G of this form is S4 [31, Theorem 1.33]. The group S4 has a versal action on P2 by the proof of Theorem 4.5. It remains to consider J = 1. We shall see that this case cannot occur. Suppose J contains a subgroup M π −1 (M )  C2 × C2 .  The group M =  ⊂ G has essential dimension ≤ 2 and a non-trivial centre (it is  a 2-group). Thus, there is an embedding ρ : M → GL2 (C). This representation ρ is faithful and, since GK ⊂ Z(M ), has a non-cyclic centre. It cannot be irreducible by Schur’s lemma. Thus, M is abelian and must have a fixed point on X (since it is versal). Under the projection to B this becomes a fixed point for M . But M has rank 2 and cannot have a fixed point on B  P1 , a contradiction. Thus we  cannot have a subgroup C2 × C2 in J. We have a morphism GB → S3 whose kernel cannot contain C2 × C2 . Considering the normal structure of GB (a polyhedral group), this excludes GB isomorphic to A4 , S4 or A5 . It remains to consider GB cyclic or dihedral. The involutions in Aut(GK ) all fix a non-trivial element of GK . Since G has a trivial centre, we must have an element g ∈ G that descends to an element of order 3 in Aut(GK ). If GB is cyclic then J and π(g) generate GB . If GB is dihedral then J and π(g) generate the maximal normal cyclic subgroup of GB . Indeed, there is no non-trivial map from a dihedral group to C3 so GB surjects onto Aut(GK )  S3 . The kernel of the composition GB → Aut(GK ) → C2 is  generated by J and π(g) as desired. Note that π(g)3 ∈ J in either case. Let L = π −1 (J), g . Consider any j ∈ π −1 (J). Since π(j) and π(g) commute, we have (g, j) = k for some k ∈ GK . Thus j 2 ∈ Z(L) since gj 2 g −1 = k 2 j 2 = j 2 . Suppose J has even order. Note that Z(L) ∩ GK = 1 so there exists j ∈ π −1 (J) with j ∈ / GK such that j 2 = 1. In this case, we have a subgroup GK × j  (C2 )3 ⊂ G. This cannot have essential dimension 2 so we have  a contradiction. We may assume J has odd order. Note that Z(L) ⊂ π −1 (J) since any 39  element mapping non-trivially to L/π −1 (J) ⊂ Aut(GK ) cannot be central. We want to show that π maps Z(L) onto J. For any y ∈ J there exists x ∈ J such that x2 = y (since |J| is odd). There is a lift l ∈ L such that π(l) = x. We have l2 ∈ Z(L) and π(l2 ) = x2 = y as desired. Since GK ∩ Z(L) = 1 we have a splitting J → L with image Z(L). Thus we may identify J and Z(L) Since ed(L) ≤ 2 and J = Z(L) = 1, there is an embedding L → GL2 (C). We then compose this with the natural map GL2 (C) → PGL2 (C). Note that L/J  (GK  C3 )  A4 . Since Z(L) = J, we have a map L → PGL2 (C)  with image A4 and kernel J. Any subgroup of GL2 (C) mapping onto A4 ⊂ PGL2 (C) must have a central involution by Proposition 2.10. So J has even order; a contradiction.  40  Chapter 6  Del Pezzo Surfaces of Degree ≤4 We are finally in a position to prove Theorem 1.1. It remains only to show that groups with versal actions on del Pezzo surface of degree ≤ 4 have already been seen acting versally on the surfaces of Theorem 4.5. Indeed, the main theorem is an immediate consequence of the following: Theorem 1.2. If G is a finite group of essential dimension 2 then G has a versal action on P2 , P1 × P1 , DP6 , or DP5 . Proof. All groups G of essential dimension 2 have versal actions on minimal rational G-surfaces by Proposition 2.8. Thus, it suffices to prove that, for any minimal rational versal G-surface X, there exists a versal action on one of the 4 surfaces listed above. Recall that any minimal rational G-surface X is a del Pezzo surface or has a conic bundle structure by Theorem 2.7. Theorem 5.1 proves the theorem for surfaces with a conic bundle structure. We recall from [25, Section 6] that the only minimal rational G-surfaces of degree ≥ 5 are precisely those listed in the statement of the theorem. Thus it suffices to consider degrees ≤ 4. In the following X is a del Pezzo surface with a versal G-action. Case degree 4: The minimal groups of automorphisms of del Pezzo surfaces X of degree 4 are listed in [25, Theorem 6.9]. We know that G must be from this list and that ed(G) ≤ 2. If G is abelian or a 2-group then it acts versally on P2 . All remaining groups have abelian subgroups with ranks ≥ 3 (note that C2 × A4 contains C23 ); thus they cannot be versal. An alternative proof that does not rely directly on [25, Theorem 6.9] can be found in Section 6.1. 41  Case degree 3: The minimal groups of automorphisms of del Pezzo surfaces X of degree 3 are listed in [25, Theorem 6.14]. It suffices to consider G from this list. All groups with non-trivial centres and essential dimension ≤ 2 have versal actions on P2 by Proposition 2.5. Thus we may assume G has a trivial centre. In particular, we may eliminate all abelian groups from the list. Next, we may eliminate all groups with abelian subgroups of rank ≥ 3 since they cannot be versal by Proposition 2.4((b)). Similarly, we eliminate G containing a non-abelian 3-subgroup by Proposition 2.6. Also, if G is a subgroup of S5 then G has a versal action on DP5 by the proof of Theorem 4.5. All that remains to consider are G of the form C32 suffices to consider G  C2 and C32  C22 . It  C32 C22 . We may view this group as a representation  of C22 on the vector space F23 . Since the centre is trivial, we may assume the representation is faithful. The representation is diagonalisable, so G is isomorphic to S3 × S3 . This group has a versal action on P1 × P1 by the proof of Theorem 4.5. An alternative proof can be found in Section 6.1. Case degree 2: We have a finite G-equivariant morphism of degree 2 to P2 (see [25, Section 6.6]). If the induced action of G on P2 is faithful then we are done. Otherwise, the group G contains a central involution (a Geiser involution). Any such group has a non-trivial centre and sits inside GL2 (C) by Corollary 2.5. Case degree 1: This case proceeds the same way as degree 2 via the Bertini involution. The only difference is that the finite morphism of degree 2 maps onto a singular quadric cone in P3 ([25, Section 6.7]). The automorphism group of a singular quadric cone is the same as the minimal ruled surface F2 (see [28, Example V.2.11.4]). Any versal action on such a surface must also act versally on P2 or P1 × P1 (see Lemma 5.2 above).  42  6.1  Alternative Proofs  Recall that the degree 3 and 4 cases of Theorem 1.2 above used Dolgachev and Iskovskikh’s classification [25] in an essential way. The careful reader will find that the proofs in the classification are quite involved. To ease the reader’s burden, we supply alternative proofs for these two cases. Our context is considerably less ambitious than theirs, so we have the luxury of a simplified argument.  6.1.1  Del Pezzo Surfaces of Degree 4  Theorem 6.1. Suppose G is a finite group with a versal action on a del Pezzo Surface X of degree 4. Then G has a versal action on P2 or DP5 . Proof. Recall from [25] or [46] that groups acting on a del Pezzo surface of degree 4 must sit inside the Weyl group W (D5 )  C24  S5 . We see that any  group with a versal action on such a surface must also act versally on P2 or DP5 via Lemma 6.2 below. Lemma 6.2. If G is a subgroup of W (D5 )  C24  S5 of essential dimension  2 then G ⊂ GL2 (C) or G ⊂ S5 . Proof. Let K = G ∩ C24 . Clearly, K must have rank ≤ 2. If K has rank 0 then G embeds into S5 . If K has rank 1 it must be central in G, so G has a non-trivial centre and embeds in GL2 (C) by Proposition 2.5. It remains to consider K of rank 2, where K  C2 × C2 .  If 5 divides |G| then G contains an element σ of order 5. The action of σ on K must be trivial since K has fewer than 5 elements. The origin of K is the only element that can be invariant under the σ action. Thus G has no elements of order 5. Note that the automorphism group of K is S3 . Since K is a normal subgroup of G there exists a map G → Aut(K)  S3 . If an element of K is  fixed by the G-action then G has a non-trivial centre and G ⊂ GL2 (C). Thus it suffices to assume the image of G → Aut(K) must contain an element of order 3. Since S5 has no elements of order 9, we may assume there exists σ ∈ G of order 3 which acts transitively on the non-trivial elements of K. 43  It is useful to view V = C24 as a vector subspace a1 +a2 +a3 +a4 +a5 = 0 for (a1 , a2 , a3 , a4 , a5 ) ⊂ F52 with S5 permuting basis elements in the obvious way. We may choose our coordinates so that σ → (123) under the map G → G/K ⊂ S5 . Let x = (a1 , a2 , a3 , a4 , a5 ) ∈ K \ {0}, we see that y = x + xσ + xσ  2  = (a1 + a2 + a3 , a1 + a2 + a3 , a1 + a2 + a3 , a4 , a5 ). Note that y must be in K. If y = 0 then a4 = a5 = 0 and a1 + a2 + a3 = 0. The only non-trivial K satisfying these conditions is (1, 1, 0, 0, 0), (0, 1, 1, 0, 0) . The stabiliser of this group in S5 is of the form S3 × C2 with S3 permuting the first 3 basis vectors and C2 swapping the last 2. The action of S3 on (1, 1, 0), (0, 1, 1) ⊂ F32 yields a group isomorphic to S4 . We see then that G is a subgroup of S4 × C2 . If the restriction to G of the projection map S4 × C2 → S4 is an isomorphism then G ⊂ S4 ⊂ S5 ; otherwise G has a non-trivial centre and is in GL2 (C). 2  If y = 0 then y = xg for some g ∈ σ . Thus x + xg + xg = xg and 2  then x + xg = 0. Thus a1 = a2 = a3 . There are only 3 such elements in V : (0, 0, 0, 1, 1), (1, 1, 1, 1, 0) and (1, 1, 1, 0, 1). All of these elements are fixed by σ so this case does not occur.  6.1.2  Del Pezzo Surfaces of Degree 3  The finite groups acting on del Pezzo surfaces of degree 3 were classified by Segre [55]. Unfortunately, Segre’s classification had errors as pointed out by more recent classifications by Dolgachev and Iskovskikh [25], and by Hosoh [29]. However, even the newer classifications are long and technical (this is probably inevitable). Given the checkered history of this problem, it seems wise to have an independent argument in our more modest context. In addition, many of our intermediate results are stronger than the classification alone. These may be useful for extending the results to more general fields.  44  Theorem 6.3. Suppose G is a finite group with a versal action on a del Pezzo surface X of degree 3. Then G has a versal action on P2 , P1 × P1 or DP5 . Proof. We recall some basic facts from [25, Section 6.5]. Any del Pezzo surface X of degree 3 has a G-equivariant embedding X → P3 as a non-singular cubic hypersurface obtained via the anti-canonical sheaf −KX . As in [33, §4], we have a representation ρ of G acting on the 4-dimensional space V = Γ(X, −KX )  C4 which lifts the action on P3 .  Let F ∈ C[x1 , x2 , x3 , x4 ] = C[V ] be the homogeneous form of degree 3 corresponding to X in P3 . The non-singular condition puts some restrictions on the possibilities for F: Lemma 6.4. For every xi , there must be a monomial in F such that xi has exponent ≥ 2. Proof. This is proved in [25]; we include it here for the sake of completeness. Assume the contrary, that F does not have a monomial where xi occurs with exponent ≥ 2. Without loss of generality, we may assume it is x1 . The polynomial F can be written in the form x1 G(x2 , x3 , x4 ) + H(x2 , x3 , x4 ) where G and H are forms of order 2 and 3 respectively. The point (1 : 0 : 0 : 0) ∈ X is singular; a contradiction. For the rest of the proof we assume that G has a versal action on X. The remainder of the proof is structured as follows. First, we show that, if ρ is reducible, G must act versally on P2 or P1 ×P1 (Lemma 6.5). Next, we prove that if G has an element of order 5 it must act versally on DP5 (Lemma 6.6). Of the remaining groups, we prove that there are no irreducible monomial versal actions (Lemma 6.7). Finally, we eliminate all irreducible actions (Lemma 6.8). Lemma 6.5. If ρ is reducible then G has a versal action on P2 or P1 × P1 . Proof. If G has a non-trivial centre then G ⊂ GL2 (C) and we have a versal action on P2 by Proposition 2.5. Thus we may assume Z(G) = 1. Let 45  ρ = σ1 ⊕ σ2 be a decomposition into subrepresentations where dim(σ1 ) ≥ dim(σ2 ) and σ2 is irreducible. Consider the case where dim(σ2 ) = 1. Suppose there exists an element g ∈ G such that σ1 (g) = 1 but σ2 (g) = 1. Any such g would be in the centre of G, so no such elements exist. This means that σ1 is faithful. We have a 3 dimensional faithful representation of G with a trivial centre. Thus, G acts versally on P2 by Theorem 3.2. We are left with the case where σ1 and σ2 are both irreducible of dimension 2. Since the centre of G is trivial, G descends to a versal action on P1 × P1 by Theorem 3.2. Recall that we have an embedding G → W (E6 ) where W (E6 ) is the Weyl group associated to the root system E6 . Thus |G| divides |W (E6 )| = 51840 = 27 · 34 · 5. We handle the case where G has an element of order 5: Lemma 6.6. If G has an element of order 5 then G has a versal action on P2 or DP5 . Proof. Any such group G must act on the surface of type II in the table from [25, Section 6.5]. The automorphism group of this surface is S5 . So, in fact we have the stronger result that G ⊂ S5 regardless of whether G is versal. Since the proof in [25] involves complicated calculations, we sketch an alternate proof here using the fact that G is versal. The group G must be isomorphic to a subgroup of W (E6 ). The maximal subgroups of W (E6 ) with order divisible by 5 are C24 Theorem 6.2, G ⊂ C24  S5 and S6 × C2 . By  S5 implies that G ⊂ S5 or G has a non-trivial  centre. In the case where G ⊂ S6 × C2 , if the projection S6 × C2 → S6 is not injective when restricted to G then G has a non-trivial centre (and so acts versally on P2 ). Thus, it remains to consider G ⊂ S6 . The group S6 has essential dimension 3, so G cannot be the whole group. All maximal subgroups of S6 of order divisible by 5 are isomorphic to A6 or S5 . All faithful representations of A6 have dimension ≥ 5 so this group does not occur either. All maximal subgroups of A6 of order divisible by 5 are isomorphic to A5 . Thus G ⊂ S5 and has a versal action on DP5 .  46  There exists a versal cubic surface realising an S5 -action appearing in the lemma above (see [38]). All remaining cases have |G| coprime to 5 and ρ irreducible. We will show that none of the remaining cases can occur. We consider the case where ρ is an irreducible monomial action: Lemma 6.7. If |G| is coprime to 5 then ρ cannot be irreducible monomial. Proof. Since ρ is irreducible, all central elements are scalar matrices and act trivially on X. Thus, we may assume G has a trivial centre. The monomial representation on V gives us an extension 1 → K → G → H → 1 where K is an abelian group and H is a subgroup of S4 . We may assume x1 , . . . , x4 is a monomial basis for V ∗ on which K acts by diagonal matrices and H acts by “twisted” permutations (see Section 3.2). We require that H is a transitive subgroup of S4 or else ρ is reducible. Also, we require that K is non-trivial or else ρ is reducible (since all irreducible representations of S4 are of dimension ≤ 3). Let χi be the character corresponding to the action of K on xi for each i = 1, 2, 3, 4. Case 1: χi are not distinct. Without loss of generality, we may assume χ1 = χ2 . By transitivity of H, this means that χ3 = χ4 . The group H permutes the subsets {1, 2} and {3, 4}. The possible H are all subgroups of the following group isomorphic to D8 : (1 2), (1 3 2 4) . Note that H is a 2-group, so the set of odd elements Go of G is contained in K. Indeed, since K is finite abelian, Go is a subgroup of G. Also, Go is normal. The quotient map G → G/Go has a section since |Go | and |G/Go | are relatively prime. Thus there exists a 2-Sylow subgroup P of G which maps onto H. Note that G = P K. Recall that any normal subgroup of a 2-group must intersect the centre, so if |K| is even then K must contain an element s in the centre of P . Thus s is in the centre of G; a contradiction. We may assume |K| is odd. Since H is transitive, h = (1 2)(3 4) is an element of 47  H. Note that h is central in H and acts trivially on K. Since |K| and |H| are coprime, G splits as K  H. Thus h lifts to a central involution in G; a  contradiction. Case 2: all χi are distinct. Suppose F contains a monomial of the form x3i . F must contain all monomials of the form x3i by transitivity of H. Since all χi are distinct, the only fixed points of K acting on P(V ) are {(1 : 0 : 0 : 0), (0 : 1 : 0 : 0), (0 : 0 : 1 : 0), (0 : 0 : 0 : 1)}. Since x31 is the only degree 3 monomial non-zero on (1 : 0 : 0 : 0), this point is not on the surface X. Similarly, none of the other points are on X. Thus we have an abelian subgroup K without fixed points on X. This contradicts Proposition 2.3. We may assume F contains no monomials of the form x3i . From Lemma 6.4, for every i ∈ 1, 2, 3, 4 the polynomial F must contain a monomial where xi has power ≥ 2. Thus, F contains a monomial of the form x2i xj for i = j. If F also contains the monomial xi x2j then 2χi +χj = χi +2χj and we find χi = χj , a contradiction. In particular, no permutation of H contains (i j) in its cycle decomposition. If F contains a monomial x2i xk for k = j then 2χi + χj = 2χi + χk and we find χj = χk , a contradiction. Thus, we may assume there is exactly one monomial of the form x2i xj for any particular i. In particular, any element of H leaves i fixed if and only if it leaves j fixed. Without loss of generality, we may assume that x21 x2 is in F . Since H is transitive, we may assume there is an involution h ∈ H which does not fix {1}. This element cannot contain (1 2) in its cycle decomposition and if it moves {1} it must move {2}. The only possibilities are h = (1 3)(2 4) or h = (1 4)(2 3). Without loss of generality, we take h = (1 3)(2 4). There must also be x22 xj , x23 xk , and x24 xl in F for some j, k, l. With the considerations above, the only possible monomials are x22 x3 , x23 x4 , and x24 x1 . Since all of these monomials are present: 2χ1 + χ2 = 2χ2 + χ3 = 2χ3 + χ4 = 2χ4 + χ1 . 48  Combining adjacent equalities we have χ2 = 2χ1 − χ3 , χ3 = 2χ2 − χ4 and χ4 = 2χ3 − χ1 . Substituting into the second equation we find: 5χ3 = 5χ1 . Since K is an abelian group of order coprime to 5 this means χ1 = χ3 . This contradicts our assumption that the χi are distinct. The remaining cases can all be reduced to the monomial case: Lemma 6.8. If |G| is coprime to 5 then ρ cannot be irreducible. Proof. We will show that any irreducible ρ must be also be monomial and so is impossible by Lemma 6.7 above. Recall that ρ cannot contain any scalar matrices. Since ρ is irreducible, this means G has a trivial centre. It suffices to prove the theorem for G where the restricted representations of all subgroups are reducible. Since we are assuming |G| = 2a 3b , we see that G is solvable by Burnside’s theorem. Thus, we may assume G has a non-trivial normal subgroup H whose restricted representation ρ|H is reducible. From [57, Proposition 24], ρ|H is isotypic or there exists a group M containing H, and an irreducible representation σ of M such that ρ is induced by σ. Case 1: ρ|H is isotypic. If ρ|H is a direct sum of linear characters then all its elements are scalar matrices. Since G has no scalar matrices, H must be trivial: a contradiction. Thus, ρ|H is a direct sum of 2 copies of a 2-dimensional irreducible representation σ. If ρ|H has a non-trivial centre then G has a non-trivial centre. All 2groups have non-trivial centres, so H is not a 2-group. There exists an element h ∈ H of order 3. Since h is not central, σ(h) has distinct eigenvalues. Thus, ρ(h) has 2 eigenvalues, each with multiplicity 2. We may apply Lemma 6.9 below to see that ρ is monomial. Case 2: G is induced from (M, σ). We either have dim(σ) = 1 or dim(σ) = 2. If dim(σ) = 1 then ρ is monomial. The only remaining case is where dim(σ) = 2. We see that M has index 2 in G, so M is a normal subgroup of G. We may write ρ|M = σ ⊕ σ for a 2-dimensional representation σ . Let K be the 49  group consisting of all elements k ∈ M such that both σ(k) and σ (k) are scalar matrices. Suppose σ is not monomial. From Lemma 4.1 and Theorem 3.2, we see that K is non-trivial. Suppose K has an element k of order 3. Either k is central in G or we may apply Lemma 6.9 to see that ρ is monomial. In either case we have a contradiction. It remains to consider K a non-trivial abelian 2-group. Since G has a trivial centre and K is normal in G, there must be at least two distinct involutions in K. Thus K has rank at least 2 and so there is some k ∈ K such that ρ(k) = (− id2 , − id2 ). This contradicts G having a trivial centre. Thus, σ must be monomial and, so, ρ is monomial. It remains only to prove the technical lemma used above: Lemma 6.9. Suppose there exists an element g ∈ G such that g 3 = 1 and ρ(g) has 2 distinct eigenvalues, each with multiplicity 2. Then ρ is monomial. Proof. This theorem is essentially the A2 case of [25, Theorem 6.10]. Any surface with such a g must be isomorphic to the “Fermat cubic”. All group actions on the Fermat cubic lift to monomial representations on V . This proof is buried among other calculations so we sketch it here: We see that ρ(g) must have eigenvalues ζ3a and ζ3b , with a = b, and the corresponding eigenspaces are 2-dimensional. Let us choose coordinates C[x1 , x2 , x3 , x4 ] for the linear variety on which g acts and let {x1 , x2 } generate one eigenspace and {x3 , x4 } the other. When g is considered as an element of PGL4 (C), it suffices to assume a = 0 and b = 1. Consider any monomial xa11 xa22 xa33 xa44 in F . Note that g(F ) = ζ3a3 +a4 F , so a3 +a4 is constant modulo 3 for all monomials in F . If a3 +a4 ≡ 1 mod 3 then x3 , x4 never occur with an exponent ≥ 2. Similarly, if a3 + a4 ≡ 2 mod 3 then x1 , x2 never occur with an exponent ≥ 2. Both of these are impossible by Lemma 6.4. The only possibility is a3 + a4 ≡ 0 mod 3. This means that all monomials in F are of the form xa11 xa22 or xa33 xa44 .  50  Thus, F is of the form F1 (x1 , x2 ) = F2 (x3 , x4 ) for homogeneous forms F1 and F2 of degree 3. The form F1 does not have a multiple root at (a : b) (as a point in P1 ) since otherwise S would be singular at (a : b : 0 : 0). We may choose new coordinates for x1 , x2 taking the three roots to (1 : −1), (1 : −ζ3 ) and (1 : −ζ32 ) so that F1 (x1 , x2 ) = x31 + x32 . We may do the same for F2 , so we find F is a(x31 + x32 ) + b(x33 + x34 ) = 0 for a, b ∈ C× . Rescaling our variables again we may put F in the form x31 + x32 + x33 + x34 = 0. This is the Fermat cubic. Consider the subgroup M  C34 S4 of GL4 (C) generated by permutating  the variables and multiplying them by third roots of unity. This group clearly preserves the form x31 + x32 + x33 + x34 . The 27 exceptional curves on S are lines of the form x1 + ηx2 = x3 + νx4 = 0, x1 + ηx3 = x2 + νx4 = 0 or x1 + ηx4 = x2 + νx3 = 0 where η 3 = ν 3 = 1. Any automorphism of X must permute these lines. M contains all possible such automorphisms and is monomial. Thus all groups G with versal actions on a del Pezzo surface of degree 3 already act versally on P2 , P1 × P1 or DP5 . This concludes the proof of Theorem 6.3.  51  Chapter 7  Essential Dimensions of A7 and S7 In this chapter, k is a field of characteristic 0. Our goal is to prove the following: Theorem 1.3. edk (A7 ) = edk (S7 ) = 4. Buhler and Reichstein [12] establish bounds for symmetric groups when n ≥ 5: n − 3 ≥ edk (Sn ) ≥ n/2 .  (7.1)  We note that these bounds tell us that edk (S7 ) is either 3 or 4. For the alternating groups An , they found the following bounds when n ≥ 5: n − 3 ≥ edk (An ) ≥ 2 n/4 .  (7.2)  From this edk (A6 ) is either 2 or 3. Recently, Serre found edk (A6 ) = 3 (see [58, Proposition 3.6]). Our major tool is the following theorem of Prokhorov [48, Theorem 1.5]: Theorem 7.1. Let X be a rationally connected threefold over C with a faithful action of A7 . Then X is equivariantly birationally equivalent to one of the following: (i) The subvariety of P6 , with the standard permutation A7 action, cut out by symmetric polynomials of degrees 1, 2 and 3. (ii) P3 with a linear action of A7 .  52  In Serre’s proof that edk (A6 ) = 3, he uses the Enriques-Manin-Iskovskikh classification to show that the only minimal rational A6 -surface is P2 . He then uses the Going-Down Theorem (see Proposition 2.3) to show that it is not versal. Thus edk (A6 ) = 2 and, by the bounds (7.2), it must be 3. Our proof is analogous, with the two threefolds in Theorem 7.1 playing the role of P2 . Proof of Theorem 1.3. First, we prove the theorem in the case where k = C. We have the following string of inequalities: 4 ≥ edC (S7 ) ≥ edC (A7 ) ≥ edC (A6 ) = 3 . Indeed, the first inequality follows from the bound (7.1). The second and third inequalities follow from the standard fact that edk (G) ≥ edk (H) for any subgroup H of a finite group G. Thus it suffices to prove that edC (A7 ) = 3. Suppose edC (A7 ) = 3. There exists a dominant rational A7 -equivariant map ψ : V  X from a linear A7 -variety V to a 3-dimensional A7 -variety  X. From this, X is unirational and, thus, rationally connected. We may assume that X is one of the threefolds from Prokhorov’s Theorem. Note that V has an A7 -fixed point (the origin) and X is proper. Thus all abelian subgroups of A7 have fixed points by Proposition 2.3. For each threefold, we will exhibit an abelian subgroup of A7 without fixed points on X. This leads to a contradiction and, so, edC (A7 ) = 3 as desired. Case (i): Consider A = (1 2 3), (4 5 6) , an abelian subgroup of A7 . Let ζ be a third root of unity. Consider the following points in P6 : (λ1 : λ1 : λ1 : λ2 : λ2 : λ2 : λ3 ) (1 : ζ : ζ 2 : 0 : 0 : 0 : 0) (1 : ζ 2 : ζ : 0 : 0 : 0 : 0) (0 : 0 : 0 : 1 : ζ : ζ 2 : 0) (0 : 0 : 0 : 1 : ζ 2 : ζ : 0) where λ1 , λ2 , λ3 ∈ C are not all 0. These correspond to the eigenspaces  53  of a lift of A acting on C7 . Thus these are all the A-fixed points on P6 . We claim that none of these points lie on X. For points of the first form, there are only two solutions of x1 + . . . + x7 = 0 and x21 + . . . + x27 = 0: λ1 = −1 ±  √  −7,  λ2 = −1 ∓  √  −7,  λ3 = 6  One then checks that x31 + . . . + x37 = 0 for these two points and for the remaining points. We have an abelian subgroup without fixed points — a contradiction. Case (ii): In this case A7 acts linearly on P3 and can be viewed as a subgroup of PGL4 (C). Let A = (1 2)(3 4), (1 2)(5 6) be an abelian subgroup of A7 . Let B be the inverse image of A in GL4 (C). We have the following exact sequence of groups: 1 → C× → B → A → 1 where C× is the set of scalar matrices in GL4 (C). Recall that A has a fixed point on P3 . This is equivalent to saying that the action of B (viewed as a 4-dimensional linear representation) has a 1-dimensional subrepresentation χ : B → C× . This gives us a splitting B  A × C× . In particular, B is  abelian. From [17, page 10], there are two distinct projective representations of A7 inside PGL4 (C) which are quotients of representations of the double cover 2.A7 in GL4 (C). There is only one element of order 2 in 2.A7 (namely the generator of the center). Thus any lift in B of the abelian subgroup A  C2 × C2 cannot be abelian. Thus B is not abelian — a contradiction. We have proved the theorem in the case where k = C. Now we use this  to show the general case where k is any field of characteristic 0. First, note that edk (G) ≥ edK (G) for K an algebraic closure of k (see [7, Proposition 1.5]). Next, we have edK (G) = edC (G) since K and C both contain an algebraic closure of Q (see [10, Proposition 2.14(1)]). Recalling 54  the bounds (7.1) and (7.2) we have the general theorem. Considering Theorem 1.3 and Serre’s theorem that ed(A6 ) = 3, we can improve some of the known bounds in higher dimensions. From [12, Theorem 6.5], we have that edk (Sn+2 ) ≥ edk (Sn ) + 1 for any n ≥ 1. Similarly, from [12, Theorem 6.7], we have edk (An+4 ) ≥ edk (An ) + 2 for any n ≥ 4. We have the following for n ≥ 6: n+1 , 2  n   for n even  2 n − 3 ≥ edk (An ) ≥ n−1 for n ≡ 1 mod 4 . 2     n+1 for n ≡ 3 mod 4 2 n − 3 ≥ edk (Sn ) ≥  (7.3)  (7.4)  55  Chapter 8  Conclusion 8.1  Applications  In [12] and [13], the main application of essential dimension was to determine how much a “general polynomial of degree n” can be simplified via non-degenerate Tschirnhaus transformations. It is shown in these papers that ed(Sn ) is the minimal number of algebraically independent coefficients possible for a polynomial simplified in this manner. This builds on classical results due to Klein, Hermite and Joubert. Consider a general univariate polynomial equation of degree n. One may find an expression for the solution in terms of the coefficients of the polynomial, the field operations, and compositions of some family of functions. For example, for polynomials of degree ≤ 4, one can use the family of radical functions. In a modern interpretation [24], Hilbert’s thirteenth problem is to determine the minimal number of variables s(n) required for any such family of functions (Hilbert was interested in n = 7). When one allows continuous functions, Arnol’d [1] and Kolmogorov [37] showed that s(n) = 1. When one considers algebraic functions, the question is open. The value of ed(Sn ) is an upper bound for s(n) and the best lower bound for ed(Sn ) is much lower than the best upper bounds for s(n). Thus, any improvement on estimates for s(n) or ed(Sn ) would be a significant advance. As Hilbert was interested in n = 7, Theorem 1.3 can be interpreted as a solution to a variant of Hilbert’s 13th problem. More generally, consider a finite group G. We outline the connections with essential dimension, Noether’s problem, and generic polynomials. The interested reader should see [54], [22], [36], and [34]. Noether’s problem asks whether the field k(V )G is rational over k, where 56  V is the regular representation of G. The original interest in this question was motivated by an attempt to construct field extensions of Q with Galois group G. In modern language, a positive answer means there exists a generic polynomial for G over k. When k is Hilbertian (for example, when k = Q), a generic polynomial over k yields a field extension over k with Galois group G. In fact, a generic polynomial for G over k exists when k(V )G is only retract rational over k. Retract rationality is a property of varieties weaker than rationality but stronger than unirationality (see [54]). When a versal variety X has a retract rational quotient X/G, one can construct a generic polynomial with dim(X) algebraically independent parameters. In fact, when a versal variety of minimal dimension has a rational quotient, any variety corresponding to a faithful linear representation has a retract rational quotient. Since unirational and rational coincide in dimension 2, for every group G in Theorem 1.1, there exists a versal G-variety X whose rational quotient X/G is rational. Thus, we have the following corollary: Corollary 8.1. Let k be a field of characteristic 0. If G is a finite group of generic dimension ≤ 2 over k then G is isomorphic one of the groups in the statement of Theorem 1.1.  8.2  Future Research  In both of the main results of this thesis, it was crucial to identify whether or not a G-variety was versal. Thus, we have the following recognition problem: Question 8.2. Are there good criteria for determining whether a given Gvariety is versal? Even in dimension 2 over C, the question is unanswered. Theorem 1.1 does not classify all versal minimal rational G-surfaces; it only identifies which groups appear. Indeed, different G-surfaces with the same group G may not be equivariantly birationally equivalent. For example, the del Pezzo surface of degree 5 is S5 -versal by the proof of Theorem 4.5. The 57  Clebsch diagonal cubic in P4 (cut out by x30 + x31 + x32 + x33 + x34 = 0 and x0 + x1 + x2 + x3 + x4 = 0) has a versal S5 -action by a result of Hermite, see [18], [53] and [38]. These two surfaces are not S5 -equivariantly birationally equivalent. Other examples of this phenomenon can be found for abelian groups [52], for versal actions of S4 and A5 [3], and for S3 × C2 [32]. If we vary the base field, we do not even know the groups. For algebraically closed fields of non-zero characteristic, the coarse Enriques-ManinIskovskikh classification still holds. However, the Dolgachev-Iskovskikh classification no longer applies. Furthermore, unirational surfaces are not necessarily rational in this case, so the classification may be inadequate. See [58] for related discussion. For non-algebraically closed fields of characteristic 0, we know that any group of essential dimension 2 must be in the list from Theorem 1.1 (this is immediate from [7, Proposition 1.5]). However, the problem of determining which groups appear is more complicated. It is possible that a versal Gsurface over a field k may not be defined over a subfield k while there may be another versal G-surface that is defined over k . A full classification of versal minimal rational G-surfaces would remedy this situation. In higher dimensions, the problem is significantly more difficult. First, even over C, there exist unirational varieties that are not rational. Second, there is no analog of the Enriques-Manin-Iskovskikh classification here, nor the Dolgachev-Iskovskikh classification. In fact, until Prokhorov’s paper [48], it was an open question as to whether all finite groups could be embedded into the Cremona group of rank 3 [59, 6.0]. Recently, Beauville [6] has generalised our Theorem 1.3 as follows: Theorem 8.3. If G is a finite simple group of essential dimension 3 then G is isomorphic to A6 or PSL2 (F11 ). Note, however, that the question of whether or not PSL2 (F11 ) actually has essential dimension 3 is still open. For toric varieties, the techniques of Chapter 3 should be quite effective for studying the versality condition. It is unclear how these may be generalised to other varieties. There does not seem to be any compelling reason 58  why Corollary 3.6 should only be true for toric varieties since versality is a birational invariant. One might conjecture that this theorem holds for any variety: Conjecture 8.4. Let G be a finite group acting faithfully on a variety X. Then X is G-versal if and only if, for any prime p, X is Gp -versal for a Sylow p-subgroup of G. We conclude by mentioning a promising result regarding the recognition problem for versality. In this thesis, all versal varieties are constructed as the image of a compression from a linear G-variety. Until very recently, this was essentially the only known construction for versal varieties. However, in [15, Proposition 3.3], a new construction for homogeneous spaces is discussed.  59  Bibliography [1] Vladimir Arnol d. On functions of three variables. Dokl. Akad. Nauk SSSR, 114:679–681, 1957. [2] Shinzo Bannai. Construction of versal Galois coverings using toric varieties. Osaka J. Math., 44(1):139–146, 2007. [3] Shinzo Bannai and Hiro-o Tokunaga. A note on embeddings of S4 and A5 into the two-dimensional Cremona group and versal Galois covers. Publ. Res. Inst. Math. Sci., 43(4):1111–1123, 2007. [4] Lionel Bayle and Arnaud Beauville. Birational involutions of P2 . Asian J. Math., 4(1):11–17, 2000. Kodaira’s issue. [5] Arnaud Beauville. p-elementary subgroups of the Cremona group. J. Algebra, 314(2):553–564, 2007. [6] Arnaud Beauville. On finite simple groups of essential dimension 3, 2011. arXiv:1101.1372v2 [math.AG]. [7] Gr´egory Berhuy and Giordano Favi. Essential dimension: a functorial point of view (after A. Merkurjev). Doc. Math., 8:279–330 (electronic), 2003. [8] J´er´emy Blanc. Finite abelian subgroups of the Cremona group of the plane. C. R. Math. Acad. Sci. Paris, 344(1):21–26, 2007. [9] Hans F. Blichfeldt. Finite Collineation Groups. The University of Chicago Press, 1917. [10] Patrick Brosnan, Zinovy Reichstein, and Angelo Vistoli. Essential dimension and algebraic stacks. arXiv:math/0701903v1, 2007. 60  [11] Kenneth S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982. [12] Joe Buhler and Zinovy Reichstein. On the essential dimension of a finite group. Compositio Math., 106(2):159–179, 1997. [13] Joe Buhler and Zinovy Reichstein. On Tschirnhaus transformations. In Topics in number theory (University Park, PA, 1997), volume 467 of Math. Appl., pages 127–142. Kluwer Acad. Publ., Dordrecht, 1999. [14] Huah Chu, Shou-Jen Hu, Ming-Chang Kang, and Jiping Zhang. Groups with essential dimension one. Asian J. Math., 12(2):177–191, 2008. [15] Jean-Louis Colliot-Th´el`ene, Boris Kunyavski˘ı, Vladimir L. Popov, and Zinovy Reichstein. Is the function field of a reductive lie algebra purely transcendental over the field of invariants for the adjoint action?, 2009. arXiv:0901.4358v5 [math.AG]. [16] Jean-Louis Colliot-Th´el`ene and Jean-Jacques Sansuc. La descente sur les vari´et´es rationnelles. II. Duke Math. J., 54(2):375–492, 1987. [17] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson. Atlas of finite groups. Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray. [18] Daniel F. Coray. Cubic hypersurfaces and a result of Hermite. Duke Math. J., 54(2):657–670, 1987. [19] David A. Cox. The homogeneous coordinate ring of a toric variety. J. Algebraic Geom., 4(1):17–50, 1995. [20] Harold S. M. Coxeter. Regular complex polytopes. Cambridge University Press, Cambridge, second edition, 1991. [21] Tommaso de Fernex. On planar Cremona maps of prime order. Nagoya Math. J., 174:1–28, 2004.  61  [22] Frank R. DeMeyer. Generic polynomials. J. Algebra, 84(2):441–448, 1983. [23] Ulrich Derenthal. Universal torsors of del Pezzo surfaces and homogeneous spaces. Adv. Math., 213(2):849–864, 2007. [24] Jacques Dixmier. Histoire du 13e probl`eme de Hilbert. In Analyse diophantienne et g´eom´etrie alg´ebrique, volume 3 of Cahiers S´em. Hist. Math. S´er. 2, pages 85–94. Univ. Paris VI, Paris, 1993. [25] Igor V. Dolgachev and Vasily A. Iskovskikh. Finite subgroups of the plane Cremona group, 2006. arXiv:math/0610595v4 [math.AG]. [26] William Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. [27] Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre. Cohomological invariants in Galois cohomology, volume 28 of University Lecture Series. American Mathematical Society, Providence, RI, 2003. [28] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. [29] Toshio Hosoh. Automorphism groups of cubic surfaces. J. Algebra, 192(2):651–677, 1997. [30] Yi Hu and Sean Keel. Mori dream spaces and GIT. Michigan Math. J., 48:331–348, 2000. Dedicated to William Fulton on the occasion of his 60th birthday. [31] I. Martin Isaacs. Finite group theory, volume 92 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008. [32] V. A. Iskovskikh. Two nonconjugate embeddings of the group S3 × Z2 into the Cremona group. Tr. Mat. Inst. Steklova, 241(Teor. Chisel, Algebra i Algebr. Geom.):105–109, 2003.  62  [33] Vasily A. Iskovskikh. Minimal models of rational surfaces over arbitrary fields. Izv. Akad. Nauk SSSR Ser. Mat., 43(1):19–43, 237, 1979. English translation: Math. USSR-Izv. 14 (1980), no. 1, 17–39. [34] Christian U. Jensen, Arne Ledet, and Noriko Yui. Generic polynomials, volume 45 of Mathematical Sciences Research Institute Publications. Cambridge University Press, Cambridge, 2002. Constructive aspects of the inverse Galois problem. [35] Nikita A. Karpenko and Alexander S. Merkurjev. Essential dimension of finite p-groups. Invent. Math., 172(3):491–508, 2008. [36] Gregor Kemper and Elena Mattig. Generic polynomials with few parameters.  J. Symbolic Comput., 30(6):843–857, 2000.  Algorithmic  methods in Galois theory. [37] Andrey Kolmogorov. On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR, 114:953–956, 1957. [38] Hanspeter Kraft. A result of Hermite and equations of degree 5 and 6. J. Algebra, 297(1):234–253, 2006. [39] Hanspeter Kraft, Roland L¨otscher, and Gerald W. Schwarz. Compression of finite group actions and covariant dimension. II. J. Algebra, 322(1):94–107, 2009. [40] Hanspeter Kraft and Gerald W. Schwarz. Compression of finite group actions and covariant dimension. J. Algebra, 313(1):268–291, 2007. [41] Antonio Laface and Mauricio Velasco. A survey on Cox rings. Geom. Dedicata, 139:269–287, 2009. [42] Arne Ledet. Finite groups of essential dimension one. J. Algebra, 311(1):31–37, 2007.  63  [43] Martin Lorenz. Multiplicative invariant theory, volume 135 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, VI. [44] Roland L¨ otscher. Application of multihomogeneous covariants to the essential dimension of finite groups. Transform. Groups, 15(3):611–623, 2010. [45] Yuri I. Manin. Rational surfaces over perfect fields. II. Mat. Sb. (N.S.), 72 (114):161–192, 1967. [46] Yuri I. Manin. Cubic forms, volume 4 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1986. Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel. [47] Aurel Meyer and Zinovy Reichstein.  Some consequences of the  Karpenko-Merkurjev theorem. arXiv:0811.2517v1 [math.AG], 2008. [48] Yuri Prokhorov. Simple finite subgroups of the Cremona group of rank 3. arXiv:0908.0678v3 [math.AG], 2009. [49] Zinovy Reichstein. On the notion of essential dimension for algebraic groups. Transform. Groups, 5(3):265–304, 2000. [50] Zinovy Reichstein. Compressions of group actions. In Invariant theory in all characteristics, volume 35 of CRM Proc. Lecture Notes, pages 199–202. Amer. Math. Soc., Providence, RI, 2004. [51] Zinovy Reichstein and Boris Youssin. Essential dimensions of algebraic groups and a resolution theorem for G-varieties. Canad. J. Math., 52(5):1018–1056, 2000. With an appendix by J´anos Koll´ar and Endre Szab´ o. [52] Zinovy Reichstein and Boris Youssin. A birational invariant for algebraic group actions. Pacific J. Math., 204(1):223–246, 2002.  64  [53] Zinovy Reichstein and Boris Youssin. Conditions satisfied by characteristic polynomials in fields and division algebras. J. Pure Appl. Algebra, 166(1-2):165–189, 2002. [54] David J. Saltman. Generic Galois extensions and problems in field theory. Adv. in Math., 43(3):250–283, 1982. [55] B. Segre. The Non-singular Cubic Surfaces. Oxford University Press, Oxford, 1942. [56] Vera V. Serganova and Alexei N. Skorobogatov. Del Pezzo surfaces and representation theory. Algebra Number Theory, 1(4):393–419, 2007. [57] Jean-Pierre Serre. Linear representations of finite groups. SpringerVerlag, New York, 1977. Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42. [58] Jean-Pierre Serre. Le groupe de Cremona et ses sous-groupes finis. S´eminaire Bourbaki, 2008. [59] Jean-Pierre Serre. A Minkowski-style bound for the order of the finite subgroups of the Cremona group of rank 2 over an arbitrary field. Moscow Math. J, 9:193–208, 2009. [60] Hiro-o Tokunaga. Two-dimensional versal G-covers and Cremona embeddings of finite groups. Kyushu J. Math., 60(2):439–456, 2006. [61] Anders Wiman. Zur Theorie der endlichen Gruppen von birationalen Transformationen in der Ebene. Math. Ann., 48(1-2):195–240, 1896. [62] De-Qi Zhang. Automorphisms of finite order on rational surfaces. J. Algebra, 238(2):560–589, 2001. With an appendix by I. Dolgachev.  65  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0071872/manifest

Comment

Related Items