thesis

1. Ekedahl Invariants, Veronese Modules and Linear Recurrence Varieties Ivan Martino Doctoral Thesis in Mathematics at Stockholm University, Sweden 2014

2. Ekedahl Invariants, Veronese Modules and Linear Recurrence Varieties Ivan Martino

4. Ekedahl Invariants, Veronese Modules and Linear Recurrence Varieties Ivan Martino

6. Abstract The title of this thesis refers to the three parts of which it is composed. The first part concerns the Ekedahl Invariants, a family of new geometric invariants for finite groups introduced in 2009 by Torsten Ekedahl. In Papers A and B, I prove that if G is a subgroup of GL3(C), then the class of its classifying stack is trivial in the Kontsevich value ring of algebraic varieties. This implies that such groups have trivial Ekedahl invariants. If G is a subgroup of GLn(C) with abelian reduction in PGLn(C), then I show that the Ekedahl invariants satisfy a recurrence relation in a Grothendieck type structure L0(Ab). This relation involves certain cohomologies of the resolution of the singularities of the quotient scheme Pn1 C =G, where G acts canonically on Pn1 C . Finally, I prove that the fifth discrete Heisenberg group H5 has trivial Ekedahl invariants. The second part of this work focuses on the Veronese modules (Paper C). We extend the results of Bruns and Herzog (about the squarefree divisor com-plex) and Paul (about the pile simplicial complex) to the Veronese embeddings and the Veronese modules. We also prove a closed formula for their Hilbert series. Using these results, we study the linearity of the resolution, we charac-terize when the Veronese modules are Cohen-Macaulay and we give explicit examples of Betti tables of Veronese embeddings. In the last part of the thesis (Paper D) we prove the existence of linear recurrences of orderM with a non-trivial solution vanishing exactly on a subset of the gaps of a numerical semigroup S finitely generated by a1 a2 aN = M. This relates to the recent study of linear recurrence varieties by Ralf Fröberg and Boris Shapiro. ©Ivan Martino, Stockholm 2014 ISBN 978-91-7447-895-2 Printed in Sweden by US-AB, Stockholm 2014 Distributor: Department of Mathematics, Stockholm University

8. Acknowledgements I am very pleased and honored to have been given the opportunity to work with Anders Björner, Torsten Ekedahl and Ralf Fröberg: three completely different styles of doing mathematics and of being mathematicians. First, I have to thank Torsten Ekedahl. Looking backward, I can see how much I have learned from him. He has been an inspiration for me and it always will be. Then, I really thank my supervisor Anders Björner. He provided me a great support and helped me to grow as a mathematician and as a person. Last but not the least, I want to thank Ralf Fröberg. It is difficult to sum-marize all the experiences I had with Ralf: conferences, discussions, fika, ... but for all of them I really thank Ralf. I am also grateful to Angelo Vistoli who took me as one of his students during a visiting semester in Pisa. Special thanks go to Alexander Engström, Boris Shapiro, Bruno Benedetti, Jörgen Backelin, Mats Boji and Matthew Stamps, because they spent a lot of time with me and I really thank them for this. During my doctoral studies I met amazing people. My life in Stockholm started with two Alessandro’s and Carlo: we shared the first steps into a new unknown country. Then, my walk continued with Yohannes and Johan, won-derful officemates. A big hug goes also to Kerstin, very important friend. My life would be different without Alessandro (yes another one!) and Olof: they have been incredible colleagues and we shared a really warm room! Similarly, I have to thank (in random order) Afshin, Tony, Christian, Sarah, Pinar, Martina, Rune, Stefano, Ketil, Christopher, Jens, Björn and Felix. I am so deep in debt to my dear friends in Pisa: Augusto, Fabio, Federico, Iga, LaFrancesca, Marco and Simone. Moreover, I also need to mention the friends at the GIH-pool: Anna, Arianna, Bettina, Carolina, Dan, Ellinor, Jonas, Michael, Nina, Patricia, Peter, Oxana and Stefanie. In particular a thank goes to my diving friend Alexander. Finally, I specially thank Ornella. I shared and I share with her all my moments, sad or happy: you are all for me!

10. List of Papers The following papers, referred to in the text by their Capital letters, are in-cluded in this thesis. PAPER A: Introduction to the Ekedahl invariants Ivan Martino, Submitted. PAPER B: The Ekedahl Invariants for finite groups Ivan Martino, Submitted. PAPER C: Syzygies of Veronese Modules Ornella Greco, Ivan Martino, Submitted. PAPER D: On the variety of linear recurrences and numerical semigroups Ivan Martino, Luca Martino, Semigroup Forum, Vol. 87, Issue 3 (2013) DOI: 10.1007/s00233-013-9551-2.

12. Contents Abstract v Acknowledgements vii List of Papers ix 1 Introduction 13 1.1 The Ekedahl Invariants . . . . . . . . . . . . . . . . . . . . . 14 1.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 The Veronese Modules . . . . . . . . . . . . . . . . . . . . . 21 1.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 21 1.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3 The Linear Recurrence Varieties . . . . . . . . . . . . . . . . 26 1.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 26 1.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 28 References 31 2 Paper A 33 3 Paper B 49 4 Paper C 73 5 Paper D 95

14. 1. Introduction This work is a collection of four papers at the intersection of Algebraic Geo-metry, Commutative Algebra and Combinatorics. The first two concern certain new geometric invariants for finite groups due to Torsten Ekedahl. The third one discusses a combinatorial approach to the study of Veronese modules. The last one shows a recent result on linear recurrence varieties via the use of nu-merical semigroups. For this reason, the introduction is made of three parts: the Ekedahl invari-ants, the Veronese modules and the linear recurrence varieties. In each of these parts, there is a preliminary section where I try to describe informally some of the concepts I will use. I give room to historical facts, general knowledge and I set up notations. These preliminary parts should be viewed as a short handbook for the results in the papers. After these preliminaries, I conclude each section by highlighting my main results in these topics. 13

15. 1.1 The Ekedahl Invariants I begin by focusing on the Noether Problem and on the Bogomolov multiplier and I try to blaze a trail from these to the recent research developments. I give a definition of Ekedahl invariants not involving the theory of algebraic stacks, because they can be defined in terms of classical geometry and, in this version, the concept may be useful to a wider audience. All the known examples where the Ekedahl invariants are nontrivial are counterexamples to the Noether Problem, because the second Ekedahl invariant is equal to the class of the dual of the Bogomolov multiplier in a Grothendieck type structure. The study of the Ekedahl invariants is contained in works A and B. In the first work I set up all the details one needs in Paper B. I also give a non-stacky introduction to the Ekedahl invariants, because not all the technical notions of the theory of algebraic stacks are necessary to work with these new invariants. In Paper B, I study the Ekedahl invariants for the finite subgroups of GLn(C) (in particular aiming to answer Question 1 in Section 1.1.1). Then I focus on the discrete Heisenberg group Hp (see Questions 2 in Section 1.1.1). 1.1.1 Preliminaries The Noether problem Let F be a field, G be a finite group and F(xg : g 2 G) be the field of rational functions with variables indexed by the elements of the group G. One sets the G-action on F(xg : g 2 G) via h xg = xhg and so we consider the field of invariants F(xg : g 2 G)G. In 1914, Amalie Emmy Noether wondered if the field extension F F(xg : g 2 G)G is rational (i.e. purely transcendental). To fix the idea let us consider an easy example. Let F = Q and G = Z=2Z. The identity of Z=2Z acts on Q(x0;x1) by fixing both variables and the other ele-ment switches them. The extension Q(x0;x1)Z=2Z Q(x0;x1) is proper because x0 =2 Q(x0;x1)Z=2Z. One observes that x0+x1 and x0x1 are invariants generating Q(x0;x1)Z=2Z. Therefore, the extension Q Q(x0;x1)Z=2Z is rational. A positive answer to the Noether problem was conjectured until the break-through result of of Swan [17] in 1969: «... the conjecture has proved to be extremely intractable. I will show here that there is a good reason for this. The conjecture is false even in the simplest case of a cyclic permutation group ...». Indeed, he proved that the extension Q Q(xg : g 2 Z=pZ)Z=pZ is not rational for p = 47;113 and 233. After this, a lot of effort was spent on the Noether problem, but one had to wait for Saltman’s result [16] in 1984 to get a more complete picture. He 14

16. proved that for any field F and for any prime p with (charF; p) = 1, there exists a group G of order p9 such that F F(xg : g 2 G)G is not rational. To get his goal, Saltman used a cohomological invariant introduced by Artin and Mumford [1]. In [3], Bogomolov improved Saltman’s statement from p9 to p6 by show-ing a concrete way to compute this invariant, that is now called Bogomolov multiplier and denoted by B0(G): it is a cohomological obstruction to the ra-tionality of F(xg : g 2 G)G=F. In other words, the vanishing of B0(G) is a necessary condition for the rationality of F(xg : g 2 G)G=F. He proved that the Bogomolov multiplier is a subgroup of H2 (G;C), de-fined as the cohomology classes in H2 (G;C) such that their restrictions to any abelian subgroup of G is zero. This means that B0(G) = A Ker H2 (G;C))!H2 (A;C) ; where the intersection runs over the abelian subgroups A G. A question in the Kontsevich value ring Let k be an algebraically closed field of characteristic zero. The Grothendieck group of varieties K0(Vark) is the group generated by the isomorphism classes fXg of algebraic k-varieties X modulo the relation fXg = fZg+fX nZg; for every closed subvariety Z of X. If one sets fXgfYg = fX Yg; then K0(Vark) has also a ring structure. We denote by L = fA1 kg, the class of the affine line (called Lefschetz class). The Motivic ring is the localization K0(Vark)[L1]. One can also define a filtration Filn = ffXg=Li : dimX i ng: K0(Vark)[L1] The Kontsevich value ringcK0(Vark) is the completion of the Motivic ring with respect to this filtration. Let us consider a faithful k-representation V of a finite group G. Hence, G GL(V). The beginning of the story of the Ekedahl invariants is the fol-lowing natural question. Question 1. For which finite groups G does the equality fGL(V)=Gg = fGL(V)g (1.1) hold in the Kontsevich value ring cK0(Vark) of algebraic k-varieties? 15

17. Ekedahl found out that all the known examples where the above equality does not hold are counterexamples to the Noether Problem. A non-stacky definition of the Ekedahl invariants In mathematics, to study manifolds, varieties and more generally topological spaces one constructs abstract structures like the cohomology groups, the ho-motopy groups pj() for j 2 etc. These could be seen as functors from the category of our objects C to the category of Abelian groups Ab. The invariants we are going to define need a more refined target: let L0(Ab) be the group generated by the isomorphism classes fGg of finitely generated abelian groups G under the relation fABg = fAg+fBg. It is worth remark-ing that in L0(Ab) there are differences of groups that do not correspond to any group: while fZg+fZ=5g is equivalent to the class of fZZ=5g, the element fZgfZ=5g is not the class of any group. Let V be a faithful k-representation of the finite group G and let m be a positive integer. Then G acts naturally componentswise on Vm and the quotient scheme Vm=G is usually singular. It is important to remark (see [2]) that K0(Vark) is generated by the class of smooth and proper varieties subjects to the relations fXg+fEg = fBlY (X)g+fYg; where BlY (X) is the blow up of X along Y with exceptional divisor E: BlY (X) X ^ [ E ^ [ Y: Hence, if X is the compactification of a resolution of the singularities of Vm=G, one could write fVm=Gg 2 K0(Vark) as the finite sum of classes of smooth and proper varieties fXg and fXjg, fVm=Gg = fXg+åj njfXjg 2 K0(Vark). Definition 1. Let X be a smooth and proper resolution of Vm=G: X p !Vm=G: For m large enough and i 2 Z, the i-th Ekedahl invariant ei (G) is defined as: ei (G) = fH2mi (X;Z)g+åj njfH2mi (Xj;Z)g 2 L0(Ab); where fVm=Gg 2 K0(Vark) is written as the sum finite of classes of smooth and proper varieties fXg and fXjg and fVm=Gg = fXg+åj njfXjg 2 K0(Vark). 16

18. Open questions about the Ekedahl invariants The next theorem links these new invariants to the Noether problem. Theorem 1 (Theorem 5.1 in [10]). Let G be a finite group. The following holds: a) ei (G) = 0, for i 0; b) e0 (G) = fZg; c) e1 (G) = 0; d) e2 (G) = fB0(G)_g, where B0(G)_ is the dual of the Bogomolov multiplier of the group G. We say that a group G has trivial Ekedahl invariants if ei (G)=0, for i6=0. It is important to stress that the triviality of the Ekedahl invariants is related to Question 1: Lemma 1. If the equality (1.1) holds in cK0(Vark), then G has trivial Ekedahl invariants. There are certain results about the triviality of these invariants: Proposition 1. In the following cases the equality (1.1) holds: 1) if G is the symmetric group and for every field k; 2) if G GL1 and for every field k (in particular, if G is a cyclic group); 3) if G is a unipotent finite group and for every field k; 4) if G is a finite subgroup of the group of affine transformations of A1 k and for every algebraically closed field k. In particular, their Ekedahl invariants are trivial. Vice versa, to study if the Ekedahl invariants are not trivial, one uses that e2 (G) = fB0(G)_g. Indeed B0(G)6= 0 implies that F(xg : g 2 G)G=F is not rational, and therefore the given counterexamples to the Noether problem (with B0(G)6= 0) are also the first examples of finite groups with non trivial Ekedahl invariants. Corollary 1 (Non triviality). The second Ekedahl invariant is non trivial for every algebraically closed field k with char(k) = 0 and for the groups of order p9 in Saltman’s paper [16] and of order p6 in Bogomolov’s paper [3]. Another connection between the Noether problem and Question 1 is also given by the next proposition. Proposition 2 (Corollary 5.8 in [10]). fGL(V)=Z=47Zg6= fGL(V)g in cK0(VarQ). In the literature, there are no other examples of groups with non trivial Ekedahl invariants. Following this point of view, it seems natural to ask: 17

19. Question 2. Is there any group G such that e3 (G)6= 0? More generally: Question 3. Is there any group G such that ei (G)6= 0 for some i 2? 1.1.2 Results The class of the classifying stack Our discussion about the Ekedahl invariants started with Question 1, that asks about the equality fGL(V)=Gg = fGL(V)g in the Kontsevich value ring of alge-braic k-varieties cK0(Vark). Since fGL(V)g is invertible in cK0(Vark), one can consider, instead, the equality fGL(V)=Gg=fGL(V)g = 1 2cK0(Vark): The left hand side of the latter is actually the class of the classifying stack BG of the group G. The reader who is not an expert in the theory of algebraic stacks can take this as a formal notation and just continue reading from the next Lemma-Notation: no other concepts are needed. Anyway, for completeness I recall that a G-torsor P over a scheme X over k, P ! X, is a scheme with a regular G-action. Definition 2. The classifying stack BG of a group G is a pseudo-functor from the category of schemes over k, Schk, to the category of groupoids over k, Gpdk, sending an open scheme U to the groupoid of G-torsors over U: BG : Schk ! Gpdk U7! fG-torsors over Ug: Equivalently, the classifying stack of the group G is usually defined as the stack quotient BG = [=G]. Lemma-Notation. (see Proposition 2.6 in Paper A) The class of the classify-ing stack of the group G in the Kontsevich value ring is fBGg = fGL(V)=Gg fGL(V)g 2cK0(Vark): In paper B, I deal with the finite subgroups of GLn (C) and prove that: Theorem 2 (Theorem 2.5 in Paper B). If G is a finite subgroup of GL3 (C), then fBGg = 1. 18

20. This is a partial answer to Question 1. I tried also to approach the finite subgroups of GL4, but this need the study of the resolutions of singularities of C3=G. This could be done for few cases, but not with the same general approach that I used for GL3 (C). As an application of this theorem one gets: Corollary 2. If G is a finite subgroup of GL3 (C), then the Ekedahl invariants of G are trivial. This result extends item 2) in Proposition 1. A stacky definition of the Ekedahl invariants For any integer k, Ekedahl defined a cohomological map Hk :cK0(Vark)!L0(Ab); sending fXg=Lm to fHk+2m (X;Z)g, for any smooth and proper X. This is an extension of the natural map that associates to every smooth and proper k-variety X the class fHk (X)g of its integral cohomology group Hk (X). This extension is well defined and it is a continuous group homomorphism (see Theorem 3.1 in Paper B). In [10], Ekedahl defines these new geometric invariants: Definition 3. The i-th Ekedahl invariant is ei (G) = Hi (fBGg) 2 L0(Ab). In Paper A, I prove that: Theorem 3 (Proposition 4.1 in Paper A). Definition 1 and Definition 3 are equivalent. The Ekedahl invariants for finite groups We have already seen that if G is a subgroup of GL3 (C), then the Ekedahl invariants of G are trivial. Therefore one could study groups with faithful representations in higher dimension, but one should also be able to construct the resolution in Definition 1. The discrete Heisenberg group Hp fits in this description. This is the subgroup of upper unitriangular matrices of GL3 (Fp): Hp = 8 0 @ : 1 a b 0 1 c 0 0 1 1 A : a;b;c 2 Fp 9= ;: 19

21. It is also an interesting candidate for the study of ei (Hp), because B0(Hp) vanishes and so the first unknown Ekedahl invariants is e3 (Hp). Let V be a faithful linear p-dimensional complex representation of Hp. Let Ap be Hp modulo its center. Then, I show in Theorem 4.4 of Paper B that e3 (Hp) = e4 (Hp) = ftor(H2p5 (Xp;Z))g; where Xp is smooth and projective resolution of P(V)=Ap. I am able to construct X5 via toric resolutions of singularities: in Section 4.1 in Paper B I show that tor(H5 (X5;Z)) is zero. Theorem 4 (Theorem 4.4 in Paper B). The Ekedahl invariants of the fifth dis-crete Heisenberg group are trivial. I actually give a general approach for the study of the Ekedahl invariants of Hp, but we narrow down our investigation to p = 5 because of the difficulties to extend (for every p) the technical result in Theorem 4.7 in Paper B. The reduction of a finite group G in PGLn is denoted by H: 0 K G H 0 0 C _ GLn _ PGLn _ 0: If H is commutative, I give a recurrence relation that the Ekedahl invariants satisfy in L0(Ab): Theorem 5 (Theorem 3.1 in Paper B). Let G be a finite subgroup of GLn (C) and let H be the image of G under the canonical projection into PGLn(C). If H is abelian and if Pn1 C =H has only zero dimensional singularities, then for every integer k ek (G)+ek+2 (G)++ek+2(n1) (G) = fHk (X;Z)g; where X is a smooth and proper resolution of Pn1 C =H. 20

22. 1.2 The Veronese Modules In the first part of this section, I recall some definitions about resolutions of modules over a ring and I introduce some combinatorial objects, the squarefree divisor complex and the pile simplicial complex, playing an important role in the theory of semigroup rings. I also give some known facts about the Betti numbers of the Veronese embeddings. In [13] (work not included in this thesis), we gave an algebraic proof of some polynomial identities among Betti numbers of (numerical) semigroup rings. In Paper C, we deal with the Veronese embeddings and the Veronese modules. First, we extend the results of Bruns and Herzog (see Theorem 6) and Paul (see Theorem 7) to these modules. Then, we prove a closed formula for their Hilbert series. We also show several applications of these results. 1.2.1 Preliminaries Rings and their resolutions In this section, I recall some well known concepts in commutative algebra to set the notations and avoid ambiguities. Let k be an algebraically closed field with characteristic zero. From now on we denote by S the polynomial ring in n variables with degxi = 1. Let R be a finitely generated k-algebra and M be a finitely generated R-module. The depth of M, depthM, is the length of the longest possible M-sequence in R. We denote by dimM the Krull dimension of R=AnnM, where AnnM = fr 2 R : rm = 0;8m 2 Mg is the annihilator of M. One sees that depthM dimM. Definition 4. An R-module M is Cohen-Macaulay if and only if depthM = dimM. Let M = iMi be a finitely generated N-graded S-module. We recall that the Hilbert series of the module M is H(M; z) = ¥å i=0 dimMi zi: Since M is a module over the polynomial ring S = k[x1; : : : ;xn], then H(M; z) has a rational form H(M; z) = h(z) (1z)n ; where the numerator of this fraction (called h-polynomial) plays an important role in this subject. 21

23. A minimal free resolution of the S-module M is an exact sequence of free modules 0!Fm fm !Fm1 fm1 ! f1 !F0 f0 !M !0: such that the homomorphisms fi are homogeneous and minimal (that is fi(Fi) (x1; : : : ;xn)Fi1). In this case the resolution looks like 0!jS(j)bm; j(M) !jS(j)bm1; j(M) !!jS(j)b0; j(M) !M !0: Here, S(j) is a module obtained by shifting the degrees of S by j, that is S(j)l = Slj. The polynomial ring S is naturally Nn-graded. Let S = jSj. (Without any confusion with the field k, we always use bold font for vectors.) If m = xa1 xa2 1 n is a monomial in S, then we set the notation m = xa, where a = 2 xan (a1; : : : ;an) 2Nn. Similarly one could find a minimal free resolution and define the numbers bi;j(M). Definition 5. Let M = iMi be a finitely generated N-graded S-module. The i-th Betti number of M is bi(M) = åj bi; j(M). The i-graded Betti number with degree j of M is bi; j(M). Similarly, if M = jMj is Nn-graded, then the i-graded Betti number with degree j of M is bi;j(M). It is an important fact that the Betti numbers do not depend on the resolution. By the Hilbert’s syzygy theorem (see [9]), there are only a finite number of non-zero Betti numbers and it is easy to see that bi; j = 0 if i j. For this reason it is common in the literature to write down the Betti numbers in a table like the following: 0 1 : : : p 0 b0;0 b1;1 : : : bp;p 1 b0;1 b1;2 : : : bp;p+1 2 b0;2 b1;3 : : : bp;p+2 ... ... ... . . . ... i b0;i b1;1+i : : : bp;p+i ... ... ... . . . ... n b0;n b1;1+n : : : bp;p+n The number p is called the projective dimension of M, pdimM, and we have the Auslander-Buchsbaum formula (see [4; 12]): pdimM+depthM = depthR: Using the table above we state the last definition. Definition 6. The S-module M has a linear resolution if there is only one row of non-zero Betti numbers. 22

24. Semigroup rings A monoid is a set equipped with an operation having an identity element and being associative. A monoid is affine if it is a finitely generated submonoid of Zd with respect to the addition operation. An affine monoid is positive if it is isomorphic to an affine monoid inside Nd. For this reason, when we consider an affine positive monoid H = hh1; : : : ;hNi, we can always assume that H Nd. Moreover, if H N then it is called a numerical monoid (or numerical semigroup). As we said, the polynomial ring S is naturally Nn-graded, but one could use a different multi-grading by a positive affine monoid H. Indeed, a semigroup ring k[H] is the subalgebra of S defined as k[xh : h 2 H]. The ring k[H] is generated by the monomials fxhig. In other words, a presentation of k[H] is given by f : k[y1; : : : ;yN] ! k[x1; : : : ;xn] yi7! xhi : and k[H] = k[y1;:::;yN]=kerf. Thus, k[H] is also a module over k[y1; : : : ;yN]. The squarefree divisor complex and the pile simplicial complex Set [N] = f1; : : : ;Ng. An abstract simplicial complex D on [N] is a subfamily of 2[N] such that if A 2 D and B A then B 2 D. An element of D is called face. If A is a face, we set dimA = #A1 (and dim /0 = 1). Moreover dimD is the maximal dimension of its faces. The k-skeleton of D is denoted by Dhki and it is the subcomplex of D consisting of the faces of dimension less then or equal to k. Let H = hh1; : : : ;hNi be an affine positive monoid. Bruns and Herzog in [5] defined the following simplicial complex. Definition 7. Given h 2 H, the squarefree divisor complex of k[H] is Dh(k[H]) = fi1; : : : ; ikg [N]j xhi1++hik divides xh in H : They showed that one could read the graded Betti numbers of k[H] as the reduced homology of Dh(k[H]): Theorem 6 (Proposition 1.1 in [5]). bi;h(k[H]) = dimk ˜H i1(Dh(k[H]);k). Let us define the partial ordering in Zn as a b if and only if ba 2 Nn. In the same direction Paul (see [15]) defined a combinatorial version of Dh: 23

25. Definition 8. Let A be a finite subset of Nn and let #A = N. For every c 2 Zn, the pile simplicial complex of A is Gc(A) = fF Aj å a2F a cg: Paul proved that Gc(A)=Dc(k[H]) if and only if the semigroup generated by A in Nn equals the group generated by A in Zn intersected by Nn (see Proposition 3 in [15]). In Theorem 1 of [15], he also proved a duality formula for Gc(A), namely: ˜H i1(Gc(A);k) = ˜H Nni1(Gˆc(A);k)_; (1.2) where c ˆ= åa2A ac1. The Veronese modules From now on, given a vector z = (z1; : : : ; zn) 2 Zn, we denote by jzj the total degree of z, that is jzj = z1+z2++zn. Definition 9. Let Ad = fa 2 Nn : jaj = dg. The Veronese subring S(d) of S is the algebra S(d) = k[xa : a 2 Ad]. The embedding dimension, emb(S(d)), of S(d) is N = d+n1 d = #A. Definition 10. Let S = k[x1; : : : ;xn]. The Veronese module Sn;d;k is defined as Sn;d;k = i0Sdi+k with n;d;k 2 N. Let us notice that Sn;d;0 = S(d). Furthermore, we have that Sn;d;k are S(d)- modules. Finally Paul connected the homology of the pile simplicial complex with the Betti numbers of the Veronese ring. Theorem 7 (Theorem 7 in [15]). Let i 2 Z and c 2 Zn. If jcj is a multiple of d, then bi;c(S(d)) = dimk ˜H Nni1(Gˆc(A);k): Otherwise bi;c(S(d)) = 0. 1.2.2 Results In Paper C, we study the Veronese modules of the polynomial ring in n vari-ables over an algebraically close field of characteristic zero. The main result generalizes the formulas of Paul and Bruns-Herzog to the Veronese modules. 24

26. Theorem (Theorem 3.1 in Paper C). If c is a vector in Zn such that jcj=k+ jd, then i1(Gh j1 c i;k); bi;c(Sn;d;k) = dimk ˜H where Ghcj1i is the ( j1)-skeleton of Gc. Moreover, bi;c =0 when jcj6=k+ jd. As an application, we characterize combinatorially when these modules are Cohen-Macaulay: Theorem (Theorem 3.5 in Paper C). The Veronese module Sn;d;k is Cohen- Macaulay if and only if k d. Moreover if Sn;d;k is not Cohen-Macaulay, then it has maximal projective dimension, that is pdim(Snd+n1 ;d;k) = d 1. Another application is Theorem 3.8, where we show that if k d(n1) n, then the resolution of the Veronese module Sn;d;k is pure (and actually bi = bi;k+id for all i). Another key result is that we find a general way to compute the rational form of the Hilbert series of the Veronese modules: Theorem (Theorem 2.1 in Paper C). d dzH(Sn;d;k; z) = nH(Sn+1;d;k1; z). Hence, H(Sn;d;k; z) = 1 (n1)! dn1 dzn1 zk+n1 1zd : By differentiating the latter with a computer algebra program, one could get the Hilbert series for any Sn;d;k. In particular, this lets us write a closed formula for H(S3;d;k; z). 25

27. 1.3 The Linear Recurrence Varieties In this section I begin by stating the definition of a linear recurrences varie-ty recently given by Ralf Fröberg and Boris Shapiro. This relates to certain algebraic varieties generated by Schur polynomials. I introduce some open questions and conjectures which I treat later. In addition, I discuss the con-nection between the study of linear recurrence varieties and a conjecture by Conca, Krattenthaler and Watanabe. I studied Conjecture 1 and the related Question 5 (see the section below), in the beginning of my doctoral studies. Indeed, in a published work not in-cluded in this thesis [14], we treat some regular sequences of complete sym-metric polynomials extending some of the results in [6]. In article D we study the linear recurrence varieties by using the linear recurrence associated to a numerical semigroup (giving a partial answer to Question 4). 1.3.1 Preliminaries A linear recurrence equation of order k with constant complex coefficients is an equation of the form U : un+a1un1+a2un2++akunk = 0; with n k 1 and ak6= 0. We denote by Lk the space of such linear recur-rences. Obviously, Lk = Ck1C. If the roots frig of its characteristic polynomial, p(U; z) : zk +a1zk1+a2zk2++ak; are distinct then a general solution of the recurrence equation is given by un = c1rn 1 +c2rn 2 ++ckrn k ; where the ci are complex coefficients fixed by the initial constraints. Fröberg and Shapiro in [11] introduced the following object: Definition 11. Given I = fi1 i2 img N with m k, the linear recurrence variety Vk;I is the subset of Lk consisting of all linear recurrences having at least one non trivial solution vanishing at all points of I. Proposition 4 in [11] shows that Vk;I is a quasi-affine variety (which ex-plains its name). They also posed the following question: Question 4. For which pairs (k; I) is the variety Vk;I not-empty and what is its dimension? 26

28. This variety is related to various topics in mathematics. Indeed, let us consider the following map from the k-dimensional complex affine space to the mk complex matrices, Mm;k: Mk;I : Ck ! Mm;k (x1; : : : ;xk)7! 0 BBB@ xi1 1 xi1 2 : : : xi1 k xi2 1 xi2 2 : : : xi2 k ... ... . . . ... xim 1 xim 2 : : : xim k 1 CCCA : The matrix Mk;I(x1; : : : ;xk) is called a generalized Vandermonde matrix, well-known for instance in Numerical Analysis. Observation 1. If U 2 Vk;I , then there exists a non-trivial solution fung such that ui = 0 for every i 2 I. Hence, rankMk;I(r1; : : : ;rk) k. Consider Om;k Mm;k, the subset of matrices of non-maximal rank. The braid hyperplane arrangement A Ck is defined by all the hyperplanes given by xi = xj for all i6= j. Of course, if xi = xj (for some i and j) then the rank of Mk;I(x1; : : : ;xk) is strictly less than k. Definition 12. We denote by VdA k;I the localization of M1 k;I (Om;k) into Ck nAk. In Lemma 5 of [11], the authors showed that the variety VdA k;I is the zero set of mk Schur polynomials SJ(x1; : : : ;xk), where J = f j1 j2 jkg is a k-subset of I and SJ(x1; : : : ;xk) =

41. 2 : : : x j1 x j1 1 x j1 k x j2 1 x j2 2 : : : x j2 k ... ... . . . ... x jk 1 x jk 2 : : : x jk k

66. = x0 1 x0 2 : : : x0 k 2 : : : x1 x1 1 x1 k ... .... . . ... xm1 1 xm1 2 : : : xm1 k

78. : Since the codimension of Om;k inMm;k is mk+1 (see [8]) andVdA k;I is the localization of M1 k;I equals mk+1. k;I (Om;k), the expected codimension of VdA Definition 13. A pair (k; I) is A-regular if k m 2k1 and codim(VdA k;I)= mk+1. Question 5. Set i1 = 0 and gcd(i2; : : : ; im) = 1. Which pairs (k; I) are A regular? Theorem 11 in [11] shows that if m k and codim(VdA k;I) = mk+1 then i2 = 1. 27

79. If we fix (k;m) = (3;5), this leads to a conjecture posed by Conca, Krat-tenthaler and Watanabe in [6]. Hence, I = f0;1; i3; i4; i5g. If VdA 3;I has the expected codimension, then it is a complete intersection. After some compu-tations, one can see that VdA 3;I is actually generated by hi32, hi42 and hi52, where hn(x;y; z) is the complete homogeneous symmetric polynomial of de-gree n in three variables. The authors in [6] proved the only if direction of the following conjecture. Conjecture 1. The ideal (ha(x;y; z);hb(x;y; z);hc(x;y; z)) is a complete inter-section if and only if the following conditions are satisfied: • abc 0 mod 6; • gcd(a+1;b+1;c+1) = 1; • For all positive integers grater than 2 there exists d 2 fa;b;cg such that d+26 0;1 mod t. 1.3.2 Results In Paper D, we present a partial answer to Question 4 by using the theory of semigroups. Given a numerical semigroup S = ha1; a2; : : : ; aNi under the assumption gcd(a1;a2; : : : ;aN) = 1, it is well known that the number of its gaps D(a1;a2; : : : ;aN) = NnS is finite. We define (in Paper D) the linear recurrence associate to the semigroup S as the linear recurrence given by: US : gk = w1gka1 +:::+wNgkaN ; 8 k 0; (1.3) for every choice of strictly positive real numbers fwigN i=1 and with the initial standard conditions g0 = 1, gj = 0, for aN j 0. One proves that there exists a non-zero sequence fgkgk2N satisfying the linear recurrence equation US (see Lemma 2.1 in Paper D). In Lemma 2.2 in Paper D we also show that gk = 0 if and only if k is a gap of S. Theorem 8 (Theorem 2.1 in Paper D). If S = ha1; a2; : : : ; aNi and I D(a1;a2; : : : ;aN), then V(b;I)6= /0, for all b 2 S, with b aN. This result has recently been extended by Contreras-Rojas in [7]. Theorem 9. If S and I are given as above, then V(s;I)6= /0, for all s 2 S. Finally we could say something about the dimension of the linear recur-rence variety. 28

80. Theorem 10 (Corollary 2.2 in Paper D). If S = ha1; a2; : : : ; aNi and I D(a1;a2; : : : ;aN) then dim(V(aN;I)) N, that is the Krull dimension of V(aN;I) is at least N. Let us mention that the definition of the linear recurrence associated with Ni the semigroup S actually has a probabilistic interpretation: if the coefficients wi 0 satisfy å1wi = 1, then fwig defines a probability distribution. Let =Xt be a discrete random variable taking values in N, and t 2 N. We define a random walk associated to the semigroup S = ha1;a2; : : : ;aNi as Xt = Xt1+ 8 : a1 with probability w1; a2 with probability w2; ... ... aN with probability wN; starting with X0 = 0. The probability of visiting the state k is gk = ProbfXt = k for some t 2 Ng; k 2 N and their generating function, G(z) = åk gkzk, is obtained by G(z) = 1 1w1za1 :::wNzaN : 29

81. 30

82. References [1] M. ARTIN AND D. MUMFORD, Some elementary examples of unira-tional varieties which are not rational, Proc. London Math. Soc. (3), 25 (1972), pp. 75–95. 15 [2] F. BITTNER, The universal Euler characteristic for varieties of charac-teristic zero, Compos. Math., 140 (2004), pp. 1011–1032. 16 [3] F. A. BOGOMOLOV, The Brauer group of quotient spaces of linear rep-resentations, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), pp. 485–516, 688. 15, 17 [4] W. BRUNS AND J. HERZOG, Cohen-Macaulay rings, vol. 39 of Cam-bridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993. 22 [5] , Semigroup rings and simplicial complexes, J. Pure Appl. Algebra, 122 (1997), pp. 185–208. 23 [6] A. CONCA, C. KRATTENTHALER, AND J. WATANABE, Regular se-quences of symmetric polynomials, Rend. Semin. Mat. Univ. Padova, 121 (2009), pp. 179–199. 26, 28 [7] Y. CONTRERAS-ROJAS, On the variety of recurrences associated to a numerical semigroup. Personal communication concerning work in progress, April 2014. 28 [8] J. A. EAGON AND D. G. NORTHCOTT, Ideals defined by matrices and a certain complex associated with them., Proc. Roy. Soc. Ser. A, 269 (1962), pp. 188–204. 27 [9] D. EISENBUD, Commutative algebra, vol. 150 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. 22 [10] T. EKEDAHL, A geometric invariant of a finite group. arXiv:0903.3148v1, 2009. 17, 19 31

83. [11] R. FRÖBERG AND B. SHAPIRO, Vandermonde varieties and relations among schur polynomials. Personal Website of Boris Shapiro. 26, 27 [12] S. B. IYENGAR, G. J. LEUSCHKE, A. LEYKIN, C. MILLER, E. MILLER, A. K. SINGH, AND U. WALTHER, Twenty-four hours of local cohomology, vol. 87 of Graduate Studies in Mathematics, Ameri-can Mathematical Society, Providence, RI, 2007. 22 [13] N. KUMAR AND I. MARTINO, An algebraic proof for the identities for degree of syzygies in numerical semigroup, Matematiche (Catania), 67 (2012), pp. 81–89. 21 [14] , Regular sequences of power sums and complete symmetric poly-nomials, Matematiche (Catania), 67 (2012), pp. 103–117. 26 [15] S. PAUL, A duality theorem for syzygies of Veronese ideals of weighted projective space. arXiv: 1311.5653v1. 23, 24 [16] D. J. SALTMAN, Noether’s problem over an algebraically closed field, Invent. Math., 77 (1984), pp. 71–84. 14, 17 [17] R. G. SWAN, Invariant rational functions and a problem of Steenrod, Invent. Math., 7 (1969), pp. 148–158. 14 32

86. Introduction to the Ekedahl Invariants Ivan Martino Abstract In 2009 T. Ekedahl introduced certain cohomological invariants for

87. nite groups. In this work we present these invariants and we give an equivalent de

88. nition that does not involve the notion of algebraic stacks. Moreover we show certain properties for the class of the classifying stack of a

89. nite group in the Kontsevich value ring. In [5], Ekedahl studied whether the class of the classifying stack fB Gg of a group G equals the class of a point fg in the Grothendieck group of algebraic stacks. All the known examples of

90. nite groups when this does not happen are the counterexamples to the Noether problem: one wonders about the rationality of the

91. eld extension F F(xg : g 2 G)G (see [9]). To show that fB Gg6= fg, Ekedahl introduced in [4] a new kind of geometric invariants for

92. nite groups de

93. ned as the cohomology for the classifying stack of G. We denote by K0(Vark) the Grothendieck group of varieties. Let Li be the class of the ane space Ai k in K0(Vark) (so L0 = fg, the class of a point). Let cK0(Vark) be the Kontsevich value ring of algebraic k- varieties (see Section 1). We denote by L0(Ab) the group generated by the isomorphism classes fGg of

94. nitely generated abelian groups G under the relation fA Bg = fAg + fBg. For every integer k, in [5] Ekedahl de

95. nes a cohomological map Hk : cK0(Vark) ! L0(Ab) by assigning Hk(fXg=Lm) = fHk+2m (X;Z)g for every smooth and proper k- variety X (see Section 3). The class fB Gg of the classifying stack of G can be seen as an element of cK0(Vark) (see Proposition 2.6) and so one de

96. nes: Stockholm University, Department of Mathematics, email: martino@math.su.se. 1

97. De

98. nition 3.2. For every integer i, the i-th Ekedahl invariant ei (G) of the group G is Hi(fBGg) in L0(Ab). We say that the Ekedahl invariants of G are trivial if ei (G) = 0 for i6= 0. The purpose of this paper is to introduce the theory of the Ekedahl in-variants to a reader who is not used to the notion of the algebraic stacks. For this reason we also present some unpublished results form [5, 4] aiming to a complete and self contained survey of the topic. The author believes that one could work with Ekedahl invariants with basic knowledge of algebraic geometry and for this reason we present the following non-stacky de

99. nition: De

100. nition 4.1. Let V be a n-dimensional faithful k-representation of a

101. nite group G and let X be a smooth and proper resolution of V m=G: X ! V m=G: For m large enough, the i-th Ekedahl invariant is de

102. ned as follows: ei (G) = fH2mi (X;Z)g + X j njfH2mi (Xj ;Z)g 2 L0(Ab); where fV m=Gg 2 K0(Vark) is written as the sum P of classes of smooth and proper varieties fXg and fXjg, fV m=Gg = fXg + j njfXjg. In Proposition 4.2 we prove that the two de

103. nitions are equivalent and we use the latter to prove the following theorem due to Ekedahl (see Theorem 5.1 in [4]): Theorem 4.3. We denote by B0(G)_ the dual of the Bogomolov multiplier of the group G. If G is a

104. nite group, then a) ei (G) = 0, for i 0; b) e0 (G) = fZg; c) e1 (G) = 0; d) e2 (G) = fB0(G)_g + fZg for some integer . Item d) is related to the Noether problem (see Section 1). Ekedahl ac-tually proved a stronger version of this result since he showed that e2 (G) = fB0(G)_g. 2

105. In Section 1, after a brief historical introduction, we set all the basic no-tions and notations. In Section 2 we discuss some properties of the class of the classifying stack fB Gg and in Section 3 we de

106. ne the Ekedahl invariants as the cohomology of the classifying stack. In Section 4 we present the equiv-alent non-stacky de

107. nition. Finally, in Section 4.1, we use this de

108. nition to reprove partially Theorem 5.1 of [4]. In the end of the article we recall the state of the art of the Ekedahl invariants. Notation. In all this manuscript we work over an algebraically close

109. eld k of characteristic zero. 1 Preliminaries Let F be a

110. eld and let G be a

111. nite group. We denote by F(xg : g 2 G) the

112. eld of rational functions with variables indexed by the elements of the group G. The group acts on it via h xg = xhg. We consider the

113. eld extension F F(xg : g 2 G)G; (1) where the latter denotes the

114. eld of invariants. In 1914, Emmy Noether in [9] wondered if the

115. eld extension (1) is rational (i.e. purely transcendental). Mathematicians conjectured a positive answer to the Noether problem, until the breakthrough result of of Swan (see [12]) in 1969. He proved that the extension Q Q(xg : g 2 Z=pZ)Z=pZ is not rational for p = 47; 113 and 233. After this, Saltman in [10] proved that for every

116. eld F and for any prime p with (char F; p) = 1, there exists a group G of order p9 such that the Noether problem has negative answer. He used a cohomological invariant introduced by Artin and Mumford in [1]. Bogomolov in [3], showed a concrete way to compute this invariant that is now called Bogomolov multiplier B0(G): B0(G) = A Ker H2 (G;C)) ! H2 (A;C) ; where the intersection runs over the abelian subgroups A G. This is a cohomological obstruction to the rationality of (1), i.e. the rationality of (1) implies B0(G) = 0. Bogomolov also improved Saltman's statement from p9 to p6. 3

117. In the rest of this section we introduce some de

118. nitions which we need later. The Grothendieck group K0(Vark) of varieties over k is the group gen-erated by the isomorphism classes fXg of algebraic k-varieties X, subjected to the relation fXg = fZg + fX n Zg; for all closed subvarieties Z of X. It is possible to see that K0(Vark) has a ring structure given by fXg fY g = fX Y g. The class of the empty set f;g is also denoted with 0, the class of the point fg with 1 and the class of the ane line L = fA1 kg is called Lefschetz class. Using the multiplication operation, one gets fAn kg = Ln and fPn kg = L0 + L1 + + Ln. In [2], Bittner proves that K0(Vark) is generated by the class of smooth and proper varieties modulo the relations fXg + fEg = fBlY (X)g + fY g with BlY (X) being the blow up of X along Y with exceptional divisor E: BlY (X) X ^ [ E ^ [ Y: Therefore, with the help of compacti

119. cation and resolution of singularities one writes the class of a scheme fXg 2 K0P (Vark) as a sum of classes of smooth and proper varieties fXjg: fXg = j njfXjg, with nj 2 Z. The Motivic ring of algebraic k-varieties is K0(Vark)[L1]. We naturally de

120. ne a dimension

121. ltration Filn K0(Vark)[L1] = ffXg=Li : dimX i ng: We denote by cK0(Vark) the completion of the Motivic ring with respect to this

122. ltration. This ring is called Kontsevich's value ring. De

123. nition 1.1. A G-torsor P over a scheme X over k, P ! X, is a scheme with a regular G-action. De

124. nition 1.2. The classifying stack B G of a group G is a pseudo-functor from the category of schemes over k, Schk, to the category of groupoids over k, Gpdk, sending any open scheme U to the groupoid of G-torsors over U: B G : Schk ! Gpdk U7! fG-torsors over Ug: 4

125. Equivalently, the classifying stack of the group G is usually de

126. ned as the stack quotient B G = [=G]. 2 The class of the classifying stack Recall that we denote by K0(Vark), K0(Vark)[L1] and cK0(Vark) the Grothendieck ring, the Motivic ring and, respectively, the Kontsevich value ring of algebraic k-varieties, where k is an algebraic closed

127. eld of characteristic zero. De

128. nition 2.1. We denote by K0(Stackk) the Grothendieck group of algebraic k-stacks. This is the group generated by the isomorphism classes fXg of algebraic k-stacks X of

129. nite type all of whose automorphism group scheme are ane (shortly, algebraic k-stack of

130. nite type with ane stabilizer). The elements of this group ful

131. ll the following relations: 1. for each closed substack Y of X, fXg = fY g + fZg, where Z is the complement of Y in X; 2. for each vector bundle E of constant rank n over X, fEg = fX Ang. Similarly to K0(Vark), K0(Stackk) has a ring structure. Lemma 2.2. One has that K0(Stackk) = K0(Vark)[L1; (Ln 1)1; 8n 2 N]: Moreover, the completion map K0(Vark)[L1] ! cK0(Vark) factors through K0(Vark)[L1] ! K0(Stackk) ! cK0(Vark): Proof. The

132. rst part is proved in Theorem 1.2 of [4]. Regarding the second one, we observe that Ln1 = Ln(1Ln) is invertible in cK0(VarkP). Indeed, (1 Ln)1 = 1 + Ln + L2n + : : : and each truncation xk = k j=0 Lkn belongs to Filkn. So, the serie converges in cK0(Vark). De

133. nition 2.3. A special group G is a connected algebraic group scheme of

134. nite type all of whose torsors over any extension

135. eld k K are trivial. 5

136. Let G be a special group and let X ! Y be a G-torsor of algebraic stacks of

137. nite type over k, then fXg = fGgfY g in K0(Stackk). Moreover, if F is a G-space and Z ! Y is a F-

138. bration associated to the G-torsor X ! Y and to the action on F, then fZg = fFgfY g in K0(Stackk). Those two facts are not true for a general group G. Indeed special groups play an important role in this topic (see Proposition 1.4 of [5]). Lemma 2.4. If G is a special group and if H is a closed subgroup scheme of G, then a) fGgfB Gg = 1; b) fBHg = fG=HgfB Gg. Proof. Consider the G-torsor ! [=G]. Thus, fg = fGgf[=G]g. Moreover ! [=G] is also an H-torsor and the action of H makes G=H a H-space. There is a natural G=H-

139. bration associated, [=H] ! [=G], given by BH = G=HGB G and thus fBHg = fG=HgfB Gg. We narrow down our investigation to

140. nite groups. Lemma 2.5. If V be an n-dimensional linear representation of G, then f[V m=G]g = LnmfB Gg; (2) f[P(V )=G]g = 1 + L1 + + Ln1fB Gg: (3) Proof. From the vector bundle [V=G] ! B G and from the second property in De

141. nition 2.1, one has that f[V=G]g = LnfB Gg. Similarly, one proves the

142. rst equation. Let O be the origin of V . The natural map [V nfOg=G] ! [P(V )=G] is a Gm-torsor and this implies f[V nfOg=G]g = (L 1)f[P(V )=G]g. Moreover, f[V nfOg=G]g = (Ln 1)fB Gg = (L 1)f[P(V )=G]g. Formula (2) expresses how fB Gg is connected with f[V m=G]g. The next proposition links fB Gg to fV m=Gg. Behind this result there is the study of the dierence between f[V m=G]g and fV m=Gg in K0(V ark)[L1]. We write an element of V m as v = (v1; : : : ; vn; vn+1; : : : ; v2n; : : : ; v(k1)n+1; : : : ; vkn; vkn+1; : : : ; vm) 6

143. with k = bm=nc. In other words, we consider v 2 V m as a sequence of sets made by n vectors each. Let U be the subset of V m such that at least one of the sets fvjn+1; : : : ; vjn+ng is a basis for V . We denote by M the complement of U in V m. This is a closed subset of V , because it is de

144. ned by k equations det(vjn+1; : : : ; vjn+n) = 0, for j = 0; : : : ; k 1. Therefore, codim(M) = codim(M=G) = k. We also observe that U is GLn (k)-invariant, because any linear transformation in GLn (k) moves a basis of V into another one. Moreover, GLn (k) (and so G) acts freely on it, hence [U=G] = U=G. The dierence f[V m=G]g fV m=Gg becomes f[V m=G]g fV m=Gg = (fZg + fU=Gg) (fM=Gg + fU=Gg) = fZg fM=Gg; (4) where Z is a stack, complement of U=G in [V m=G]. Similarly to M=G, Z has codimension k because both are the complement of the same object U=G, but in two dierent environments V m=G and [V m=G] with the same dimension. The class of the dierence f[V m=G]gfV m=Gg is so determined by the class of these complements. We see, in the next proposition, how this implies that fB Gg = lim m!1 fV m=Gg Lmn 2 cK0(Vark): Proposition 2.6 (Proposition 3.1 in [5]). If V is an n-dimensional faithful linear representation of G, then a) fB Gg = fGL(V )=Gg=fGL(V )g; b) The image of fB Gg in cK0(Vark) is equal to limm!1fV m=GgLmn. Proof. The general linear group is a special group and we apply Lemma 2.4.b for G GL(V ): fB Gg = fGL(V )=GgfB GL(V )g. Using Lemma 2.4.a, one gets fB GL(V )g = 1=fGL(V )g and so, we prove the

145. rst point. Using formula (2) and formula (4) one has fB Gg fV m=GgLmn = (f[V m=G]g fV m=Gg) Lmn = (fZg fM=Gg) Lmn where Z and M=G are respectively the complement of U=G

146. rstly seen inside of [V m=G] and then inside of V m=G. The open set U was de

147. ned just before this lemma. 7

148. Remark that Filj(K0(Vark)[L1]) = ffXg=Li : dimX i jg. Then, fM=GgLmn belongs to Filj(K0(Vark)[L1]) if and only if dimM=Gmn j. One knows that dimM=Gmn = codim(M=G) = k. Thus fM=GgLmn be-longs to Filj for any j k = bm=nc. Therefore, limm!1fM=GgLmn = 0 and, with a similar argument, limm!1fZgLmn = 0. Thus, fB Gg fV m=GgLmn converges to zero in cK0(Vark). 3 The Ekedahl invariants for

149. nite groups We want to de

150. ne certain cohomological maps Hk for cK0(Vark). These (and the invariants we are going to de

151. ne) need a more re

152. ned target: Let L0(Ab) be the group generated by the isomorphism classes fGg of

153. nitely generated abelian groups G under the relation fABg = fAg+fBg. We equip L0(Ab) with the discrete topology. For clari

154. cation, fZg and fZ=png belong to L0(Ab) and there are elements in L0(Ab) that do not correspond to any group: while fZg + fZ=5g is the class of fZ Z=5g, the element fZg fZ=5g is not the class of any group. It is natural to de

155. ne a cohomological map Hk : K0(Vark) ! L0(Ab); by assigning to every smooth and proper k-variety X the class of its in-tegral cohomology group Hk (X;Z). Next theorem shows that this map is well de

156. ned and that it can be extended to cK0(Vark) sending fXg=Lm to fHk+2m (X;Z)g for any smooth and proper variety X, Hk : cK0(Vark) ! L0(Ab): Theorem 3.1. The following cohomological map H : cK0(Vark) ! L0(Ab)((t)) fY g7! X k2Z Hk (fY g) tk: is well de

157. ned. For each k 2 Z, Hk : cK0(Vark) ! L0(Ab) is also a continuous group homomorphism. 8

158. Proof. The proof is given by Ekedahl in [5] via Proposition 3.2.i), ii) and Proposition 3.3.ii). We give an alternative proof. We

159. rst prove that the map Hk : K0(Vark) ! L0(Ab) is well de

160. ned. ~We know that K0X ~X (Vark) is generated by the class of smooth and proper varieties modulo the relations fXg + fEg = f g + fY g with being the blow up of X along Y (smooth subvariety of codimension d) with exceptional divisor E (note that E is also smooth because it is a projective bundle over Y , r = jE : E ! Y ): ~X X ^ h [ E ^ j [ r Y: Moreover, by the Leray-Hirsch Theorem (see for instance [7]), Hk (E) = Hk (Y ) Hk2 (Y ) Hk2(d1) (Y ) : We want to show that fHk( ~X )g + fHk (Y )g = fHk (E)g + fHk (X)g and therefore it is enough to show that Hk( ~X ) = Hk (X) Hk2 (Y ) Hk2(d1) (Y ). Firstly we observe that the pushforward of the fundamental class of ~Xis the fundamental class of X, [ ~X ] = [X] (see [6]). Now, 1 is the dual of [ ~X ] ~X and, respectively, of [X], 1 = 1. Using this and the projection formula one gets that for every y in Hk (X) (1 y) = (1) y, that is y = y and so = idHk(X). Therefore, : Hk( ) ! Hk (X) is surjective and one constructs the isomorphism Hk( ~X ) = Hk (X) ker() sending x in Hk( ~X ) into (x; x x). Calling U = X n Y we also have the following commutative diagram: : : : Hk1( ~X ) Hk1 (U) Hk2 (E) h Hk( ~X ) Hk (U) : : : _ : : : Hk1 (X) id _ Hk1 (U) r _ Hk2 (Y ) _ Hk (X) j id _ Hk (U) : : : Firstly we observe that h : ker(r) ! ker() is an isomorphism. Indeed let x be in ker(). Since x = 0, then jx = 0, but the diagram commutes and x is also the kernel of Hk( ~X ) ! Hk (U) and hence, there exists in 9

161. Hk2 (E) mapping to x. It is easy to see that belongs to ker(r). Thus the map is surjective. It is also injective because if hx = 0 then there exist

162. in Hk1 (U) mapping to x. The diagrams commutes and so in the second lines,

163. maps to zero and, hence, there exists z in Hk1 (X) mapping to

164. . The map is surjective and so there exists z0 in Hk1( ~X ) mapping to

165. in the

166. rst rows. Thus

167. has to map to zero and so x = 0. Finally we observe that ker(r) is exactly Hk2 (Y ) Hk2(d1) (Y ). This shows that Hk : K0(Vark) ! L0(Ab) is well de

168. ned. Finally, one extends this map,

169. rst to the Motivic ring and then to cK0(Vark). Without confusion we denote by 1, the class of a point fg in cK0(Vark) and we also denote by 1 = fZg 2 L0(Ab). With this notation, H (1) = 1. We

170. nally state the main de

171. nition. De

172. nition 3.2. The i-th Ekedahl invariant of a

173. nite group G is ei (G) = Hi (fB Gg) 2 L0(Ab): We say that ei (G) are trivial if ei (G) = 0 for i6= 0. 4 A non-stacky de

174. nition In this section we present an equivalent de

175. nition of these invariants that do not involve the concept of algebraic stacks. Let V be a faithful representation of a

176. nite group G. The group G acts component-wise on V m. Consider the quotient scheme V m=G that is usually a singular scheme. De

177. nition 4.1. Let V be a n-dimensional faithful k-representation of a

178. nite group G and let X be a smooth and proper resolution of V m=G: X ! V m=G: For m large enough, the i-th Ekedahl invariant is de

179. ned as follows: ei (G) = fH2mi (X;Z)g + X j njfH2mi (Xj ;Z)g 2 L0(Ab); where fV m=Gg 2 K0(Vark) is written as the sum P of classes of smooth and proper varieties fXg and fXjg, fV m=Gg = fXg + j njfXjg. 10

180. We are going to prove that there exists a positive integer M such that for any m greater then M the i-th Ekedahl invariant ei (G) stabilizes in L0(Ab). Moreover we show that the two given de

181. nitions of Ekedahl invariants are equivalent. Proposition 4.2. De

182. nition 4.1 is well posed and equivalent to De

183. nition 3.2. Proof. We have seen in Proposition 2.6.b) that fB Gg = lim m!1 fV m=GgLmn 2 cK0(Vark): From Theorem 3.1, the map Hk is continuous and so, for m large enough, Hi (fB Gg) = Hi = H2mni (fV m=Gg;Z) ; fV m=GgLmn;Z where the shifting 2mn comes from the multiplication for Lmn. We, now, consider a compacti

184. cation and resolution of the singularities of V m=G. This allows to write fV m=Gg as a suitable sum fXg + P j njfXjg where fXg is smooth, proper and birational to V m=G, the Xj's are smooth and proper with dimension strictly less then dim(V m=G) = mn and nj 2 Z. Therefore, ei (G) = Hi (fB Gg) = fH2mni (X;Z)g + X j njfH2mni (Xj ; Z)g: This proves that the two de

185. nitions are equivalent and well posed. 4.1 The state of the art The following theorem links to the Noether problem. Theorem 4.3 (Theorem 5.1 in [4]). We denote by B0(G)_ the dual of the Bogomolov multiplier of the group G. If G is a

186. nite group, then a) ei (G) = 0, for i 0; b) e0 (G) = fZg; c) e1 (G) = 0; 11

187. d) e2 (G) = fB0(G)_g + fZg for some integer . Proof. By De

188. nition 4.1, ei (G) = fH2mni (X;Z)g + X j njfH2mni (Xj ;Z)g; where X is a smooth and proper resolution of V m=G; V is a n-dimensional faithful k-representation of a

189. nite group G and fV m=Gg is written in K0(Vark) as the sum of classes of smooth and proper varieties fXjg, fV m P =Gg = fXg + j njfXjg. Let i = 0. The only surviving cohomology is H2mn (X;Z) = Z, because dim(Xj) dim(V m=G) = mn. Thus, e0 (G) = fH2mn (X;Z)g = fZg. If i = 1, for similar reasons, e1 (G) = fH2mn1 (X;Z)g. Since X is bira-tional to V m=G, one has the inclusion k(X) ' k(V=G) k(V ) and hence X is unirational and, therefore, simply connected. Thus, using the result of Serre in [11], H2mn1 (X; Z) ' H1(X;Z) = 0 and thus e1 (G) = 0. Regarding e2 (G), one

190. rstly observes, by Poincare duality, that tor(H2mn2 (X;Z)) = tor(H3(X;Z)): Artin and Mumford have proved in [1] that tor(H3 (X;Z)) is a birational invariant and Bogomolov in [3] proved that this is exactly B0(G). Therefore, we have proved that e2 (G) = fB0(G)_g + fZg for some integer . Observation 4.4. Bogomolov proved in Theorem 1.1 of [3] that if X is smooth, proper and unirational the Brauer group Brv(K) is isomorphic to tor(H3 (X;Z)), with K = k(X). Moreover he de

191. ned Brv(G) = Brv(k(X)), where X is smooth, proper and birational to V=G with V being any generically free representation of G. Thus, in Theorem 3.1 of [3], he has proved that Brv(G) = B0(G). Using these results, one

192. nds the

193. rst examples of group with non trivial Ekedahl invariants. Proposition 4.5 (Non triviality). The second Ekedahl invariant is non trivial for every algebraically closed

194. eld k with char(k) = 0 and for the groups of order p9 given by Saltman in [10] and of order p6 given by Bogomolov in [3]. Moreover in these cases, fBGg6= 1 in cK0(Vark). 12

195. Proof. The Bogomolov multiplier is always a

196. nite group and so if B0(G)6= 0, then e2 (G) = fB0(G)_g + fZg6= 0. Ekedahl actually proved a more precise statement. Theorem 4.6 (Theorem 5.1 of [4]). For i 0, ei (G) is the sum (with signs) of classes of

197. nite groups in L0(Ab). Proof. We refer to point e) of Theorem 5.1 in [4]. Corollary 4.7. The second Ekedahl invariant is exactly e2 (G) = fB0(G)_g, where B0(G)_ is the dual of the Bogomolov multiplier of the group G; Proof. We already proved in Theorem 4.3.d) that e2 (G) = fB0(G)_g+fZg for some integer . Using the previous theorem one gets = 0. Another connection between the Noether problem and the non-triviality of fB Gg is also the next proposition. Proposition 4.8 (Corollary 5.8 in [4]). fB Z=47Zg6= 1 in cK0(VarQ). To the authors knowledge, there are no examples in literature of

198. nite group G such that B0(G) = 0 and e3 (G)6= 0. Vice versa a lot of groups have trivial Ekedahl invariants. Theorem (Prop 3.2, Cor 3.9, Thm 4.3 in [4]). Assume one of the following cases: 1) if G is the symmetric group and for every

199. eld k; 2) if G GL1 and for every

200. eld k (in particular, if G is a cyclic group); 3) if G is a unipotent

201. nite group and for every

202. eld k; 4) if G is a

203. nite subgroup of the group of ane transformations of A1 k and for every algebraically closed

204. eld k. Then fB Gg = 1 2 cK0(Vark) and the trivial Ekedahl invariants are trivial. Recently the author has proved also the following facts: Theorem (Thm 2.5 in [8]). If G is a

205. nite subgroup of GL3(C), then fB Gg = 1 2 cK0(Vark) and it has trivial Ekedahl invariants. 13

206. The previous result extend point 2) in Theorem 4.1. Theorem (Thm 4.4 in [8]). The Ekedahl invariants of the

207. fth discrete Heisenberg group, ei (H5), are trivial. Acknowledgements I thank Angelo Vistoli for the great mathematical support in this subject. References [1] M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3), 25 (1972), pp. 75{95. [2] F. Bittner, The universal Euler characteristic for varieties of characteristic zero, Compos. Math., 140 (2004), pp. 1011{1032. [3] F. A. Bogomolov, The Brauer group of quotient spaces of linear representations, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), pp. 485{516, 688. [4] T. Ekedahl, A geometric invariant of a

208. nite group. arXiv:0903.3148v1, 2009. [5] , The grothendieck group of algebraic stacks. arXiv:0903.3143v2, 2009. [6] W. Fulton, Intersection theory, vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathe-matics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Springer-Verlag, Berlin, second ed., 1998. [7] A. Hatcher, Algebraic topology, Cambridge University Press, Cam-bridge, 2002. [8] I. Martino, The ekedahl invariants for

209. nite groups. arXiv:1312.0476, 2013. [9] E. Noether, Gleichungen mit vorgeschriebener Gruppe, Math. Ann., 78 (1917), pp. 221{229. 14

210. [10] D. J. Saltman, Noether's problem over an algebraically closed

211. eld, Invent. Math., 77 (1984), pp. 71{84. [11] J.-P. Serre, On the fundamental group of a unirational variety, J. London Math. Soc., 34 (1959), pp. 481{484. [12] R. G. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math., 7 (1969), pp. 148{158. 15

213. B

215. The Ekedahl Invariants for

216. nite groups Ivan Martino Abstract In 2009 Ekedahl introduced certain cohomological invariants of

217. nite groups which are naturally related to the Noether Problem. We show that these invariants are trivial for every

218. nite group in GL3 (C) and for the

219. fth discrete Heisenberg group H5. Moreover in the case of

220. nite linear groups with abelian projective reduction, these invariants ful

221. ll a recurrence relation in a certain Grothendieck group for abelian groups. Let V be a

222. nite dimension faithful linear representation of a

223. nite group G over an algebraically closed

224. eld k of characteristic zero. In [4], Ekedahl investigates when the equality fGL(V )=Gg = fGL(V )g (1) holds in the Kontsevich value ring cK0(Vark) of algebraic k-varieties. One of the motivations for Ekedahl investigations into cK0(Vark) was to make Motivic versions of point counting over

225. nite

226. elds as for instance can be found in [1]. All the known

227. nite groups G where this equality is not veri

228. ed are counterexamples to the Noether Problem (see [10]), that is to the rationality of the extension F(xg; g 2 G)G=F for every

229. eld F with G acting on F(xg; g 2 G) as h xg = xhg (see Corollary 1.5). In [5], Ekedahl also de

230. nes, for every integer k, a cohomological map Hk : cK0(Vark) ! L0(Ab); where L0(Ab) is the group generated by the isomorphism classes fGg of

231. nitely generated abelian groups G under the relation fABg = fAg+fBg. Let Li be the class of the ane space Ai k in cK0(Vark) (and L0 = fg is the 1

232. class of a point). To de

233. ne Hk on cK0(Vark) is enough to set Hk(fXg=Lm) = fHk+2m (X;Z)g for every smooth and proper k-variety X (see Section 3 in [8]). The class fB Gg of the classifying stack of G could be seen as an element of cK0(Vark) (see Proposition 2.6.b in [8]) and so one can de

234. ne: De

235. nition 1.2. For every integer i, the i-th Ekedahl invariant ei (G) of the group G is Hi(fB Gg) in L0(Ab). We say that the Ekedahl invariants of G are trivial if ei (G) = 0 for all integer i6= 0. In Proposition 2.6.a of [8], the author rephrases the equality (1) in terms of stacks, using the expression fB Gg = fGL(V )=Gg fGL(V )g 2 cK0(Vark): Therefore (1) holds if and only if fB Gg = 1 and, if this is the case, then the Ekedahl invariants of G are trivial, because H0(1) = fZg and Hk(1) = 0 for k6= 0. These invariants seem a natural generalization of the Bogomolov multiplier B0(G) (see [3]), because e2 (G) = fB0(G)_g (see Section 5 of [8]). Here B0(G)_ stands for the dual of B0(G)_. The Bogomolov multiplier B0(G) is an obstruction to the rationality of F(xg; g 2 G)G=F and thus, if the second Ekedahl invariant of G is not zero, then the group is a counterexample to the Noether Problem. It is not clear if this assertion is true for higher Ekedahl invariants. Moreover fB Gg = 1 implies that all the invariants are trivial, but we have no information about the converse. In this paper we work over the complex numbers and we prove that: Theorem 3.1. Let G be a

236. nite subgroup of GLn (C) and let H be the image of G under the canonical projection GLn (C) ! PGLn(C). If H is abelian and if Pn1 =H has only zero dimensional singularities, then for C every integer k ek (G) + ek+2 (G) + + ek+2(n1) (G) = fHk (X; Z)g; where X is a smooth and proper resolution of Pn1 C =H. In law dimensions, one can say more. 2

237. Theorem 2.5. If G is a

238. nite subgroup of GL3 (C), then fB Gg = 1 in cK0(VarC) and the Ekedahl invariants of G are trivial. The case when G is a

239. nite subgroup of GL4 (C) is more complicated because it involves a deep study of the resolution of singularities of the ane varieties C3=A, for a

240. nite group A GL3 (C), which are not well known a part few cases. For this reason we focus on the p-discrete Heisenberg group Hp, where we only deal with cyclic quotient singularities. Given a prime p we denote by Hp the subgroup of the upper triangular matrices of GL3 (Fp). In particular, Hp GLp (C). This is an interesting candidate for the study of the Ekedahl invariants, because B0(Hp) = 0 (using Lemma 4.9 in [3]) and so the

241. rst unknown Ekedahl invariant is e3 (Hp). Theorem 4.4. The Ekedahl invariants of the

242. fth discrete Heisenberg group H5 are trivial. We show a general approach for the study of the Ekedahl invariants of Hp, but we narrow down our investigation to p = 5 because of the diculties to extend the technical result in Theorem 4.7. To author's knowledge, there are no examples of

243. nite group G such that B0(G) = 0 (i.e. e2 (G) = 0) and e3 (G)6= 0. After a preliminary section where we review the theory of the Ekedahl invariants, in Section 2 we prove that these are trivial for all

244. nite subgroups of GL3 (C). Then, in Section 3, we study the

245. nite linear groups with abelian projective reduction and in the last section we deal with the

246. fth Heisenberg group. Notation. In all the work G is a

247. nite group. In Section 1, we work over an algebraically close

248. eld k of characteristic zero. In the rest of the paper, the ground

249. eld is C. 1 Preliminaries The Grothendieck ring of algebraic varieties K0(Vark) is the group generated by the isomorphism classes fXg of algebraic k-varieties X, subject to the relation fXg = fZg+fX nZg, for all closed subvarieties Z of X. The group K0(Vark) has a ring structure given by fXg fY g = fX Y g. Let L be the 3

250. class of the ane line. The completion of K0(Vark)[L1] with respect to the dimension

251. ltration Filn K0(Vark)[L1] = ffXg=Li : dimX i ng is called the Kontsevich value ring and denoted by cK0(Vark). From now on, we denote by fg the class of a point in cK0(Vark). We also use sometime the notation 1 = fg 2 cK0(Vark). Remark 1.1. Let G be a special group and let X ! Y be a G-torsor of algebraic stacks of

252. nite type over k, then fXg = fGgfY g in K0(Stackk) (see [5]). Moreover, if F is a G-space and Z ! Y is a F-

253. bration associated to the G-torsor X ! Y and to the action on F, then fZg = fFgfY g in K0(Stackk). We observe that the completion map from K0(Vark)[L1] to cK0(Vark) factors through the Grothendieck ring K0(Stackk) of algebraic stacks of

254. nite type over k (see Lemma 2.2 in [8]). The classifying stack of the group G is usually de

255. ned as the stack quotient B G = [=G] and, via this map, one sees the class of the classifying stack fB Gg as an element of cK0(Vark) (see Lemma 2.6.b in [8]). Using the Bittner presentation (see [2]), given a integer k Ekedahl de

256. nes in [5] a cohomological map for the Kontsevich value ring, sending fXg=Lm to fHk+2m (X;Z)g, for every smooth and proper k-variety X: Hk : cK0(Vark) ! L0(Ab) fXg=Lm7! fHk+2m (X; Z)g: In Section 3 of [8], we prove that this map is well de

257. ned. Notation. Every cohomology group (if not explicitly expressed dierently) is the singular cohomology group with integer coecients, that is Hk () = Hk (;Z). De

258. nition 1.2. For every integer i, the i-th Ekedahl invariant ei (G) of the group G is Hi(fB Gg) in L0(Ab). We say that the Ekedahl invariants of G are trivial if ei (G) = 0 for all integer i6= 0. The reason for the minus sign in the above de

259. nition is the following. 4

260. Lemma 1.3 (Thm 5.1 of [4]). If G is a

261. nite group, then ei (G) = 0 for every i 0. All the known

262. nite groups where fB Gg6= 1 are counterexamples to the Noether Problem. In [10], Noether wondered about the rationality of the extension F(V )G=F for any

263. nite group G and any

264. eld F, where V is a

265. nite dimension faithful linear representation of G. The

266. rst counterexample, Q(V )Z=47Z=Q, was given by Swan in [13]. Later more counterexamples were found: for every prime p Saltman (in [12]) and Bogomolov (in [3]) showed that there exists a group of order p9 and, respectively, of order p6 such that the extension C(V )G=C is not rational. Saltman used the second unrami

267. ed cohomology group of the

268. eld C(V )G, H2 nr(C(V )G; Q=Z), as a cohomological obstruction to the rationality. Later, Bogomolov found a group cohomology expression for H2 nr(C(V )G; Q=Z) which now takes his name and it is denoted by B0(G). To see the connection to the Noether problem we use the following result. Theorem 1.4 (Thm 5.1 of [4]). If G is a

269. nite group, then e0 (G) = fZg, e1 (G) = 0 and e2 (G) = fB0(G)_g, where B0(G)_ is the dual of the Bogo- molov multiplier of the group G. Moreover, for i 0, the invariant ei(G) is a sum (with signs) of classes of

270. nite abelian groups. Using that e2 (G) = fB0(G)_g, one proves that, for the Saltman and Bogo-molov counterexamples, the Ekedahl invariants are non-trivial and so fB Gg6= 1. In addition, fB Z=47Zg6= 1 2 cK0(VarQ) (see page 7 of [4]). Corollary 1.5. If G is one of the group de

271. ned in [12] and in [3] as counterexample to the Noether problem, then the second Ekedahl invariant is non- zero and so fB Gg6= 1. Proposition 1.6 (State of the art). Assume one of the following cases: 1) if G is the symmetric group and for every

272. eld k; 2) if G GL1 and for every

273. eld k (in particular, if G is a cyclic group); 3) if G is a unipotent

274. nite group and for every

275. eld k; 4) if G is a

276. nite subgroup of the group of ane transformations of A1 k and for every algebraically closed

277. eld k. 5

278. Then fB Gg = 1 2 cK0(Vark). Proof. See Proposition 3.2, Corollary 3.9 and Theorem 4.3 in [4]. All these results imply the triviality for the Ekedahl invariants. We underline that in this work we extend the result in item 2), because we prove that fB Gg = 1 for every

279. nite subgroup of GL3. A more complete and self contained introduction to the Ekedahl invariants can be found in [8]. 2 The

280. nite subgroups of GL3 (C) From now on we set C as the ground

281. eld. Let G be a subgroup of GLn and H be its reduction in PGLn: 0 K G H 0 _ 0 C _ GLn _ PGLn 0: In what follows we set V = Cn (as a linear representation of G). Notation. We sometime use for simplicity Pn1 for the projective space P(V ). We also denote by Idn the identity element of GLn. To get information about fB Gg, we study f[P(V )=G]g. Lemma 2.1 (Lemma 2.5 of [8]). f[P(V )=G]g = (1 + L + + Ln) fB Gg in cK0(VarC). By de

282. nition H acts on P(V ) and P(V )=G = P(V )=H. Their stack quo-tients are not isomorphic, but the classes of their stack quotients are equal in cK0(VarC). Proposition 2.2. f[P(V )=G]g = f[P(V )=H]g 2 cK0(VarC). Proof. We denote by VO = V nfOg, where O is the origin of V . Since [VO=G] ! [P(V )=G] is a C-torsor, f[VO=G]g = (L 1)f[P(V )=G]g (we use Remark 1.1). Similarly from [(VO=K)=H] ! [P(V )=H], one gets f[(VO=K)=H]g = (L1)f[P(V )=H]g. The statement follows from [(VO=K)=H] = [VO=G]. 6

283. We stress that the previous lemma and proposition holds for any algebraically closed

284. eld of characteristic zero. We now set up notations, de

285. nitions and remarks regarding the quotient of algebraic varieties by

286. nite groups. Let Y be a smooth quasi-projective algebraic variety and let A be a

287. nite group of automorphisms of Y . Let Y ! Y=A be the canonical quotient map and y be the image of y. The pseudo re ection subgroup Pseudo (G) of G GL(V ) is its subgroup generated by pseudo-re ections. The Chevalley-Shephard-Todd Theorem says that the quotient V=G is smooth if and only if G = Pseudo (G). The well known Cartan Lemma says that for all the points y of Y , the action of the stabilizer Staby(A) of y on Y induces an action of Staby(A) on the tangent space on y, TyY . Moreover the analytic germ (Y=A; y) is isomorphic to (TyY=A; O ), where O is the image of the origin O 2 TyY under the quotient map TyY ! TyY=A. An easy consequence is that for all the points y of Y , Staby(A) GLdim(Y ) and one also proves that p is a singular point of V=G, p 2 Sing (V=G), if and only if Pseudo (Stabp(G))6= Stabp(G). We are going to use also the following fact. A proof can be found, for instance, in the lemma in Section 1.3 of [11]. Lemma 2.3. Let y 2 Y=A. The germ (Y=A; y) is a simplicial toroidal singularity (i.e. locally isomorphic, in the analytic topology, to the origin in a simplicial toric ane variety) if and only if the quotient Staby(A)=Pseudo(Staby(A)) in TyY is abelian. Comparing the classes f[P(V )=H]g and fP(V )=Hg, we are going to prove that the Ekedahl invariants for every

288. nite subgroup G in GL3 (C) are trivial. We prove it by induction and the base of such induction is the item 2) in Proposition 1.6: if G is a

289. nite subgroup of GL1 (C) then fB Gg = 1 in cK0(VarC). Proposition 2.4. If G is a

290. nite subgroup of GL2 (C) then fB Gg = 1 in cK0(VarC) and the Ekedahl invariants of G are trivial. Proof. Let U be the open subset of P1 where H acts freely. Then f[P1=H]g = fU=Hg + X p f[p=Stabp(H)]g = fU=Hg + X p fB Stabp(H)g; 7

291. where the sum runs over the points with non trivial stabilizer. Similarly fP1=Hg = fU=Hg + X p fg and so f[P1=H]g = fP1=Hg + X p (fB Stabp(H)g fg): Using (in order) that f[P1=G]g = (1 + L) fB Gg, Proposition 2.2, the previ-ous formula and P1=H = P1, one has fB Gg(1 + L) = fP1g + X p (fB Stabp(H)g fg): Using Cartan's Lemma, Stabp(H) is a subgroup of GL1 and, hence, for Propo-sition 1.6.2), fB Stabp(H)g = fg for every non trivial stabilizer point p. Hence, fB Gg(1 + L) = fP1g and this implies fB Gg = 1, because Ln 1 is invertible in cK0(Vark), L2 1 = (L 1)(L + 1) and so 1 + L is invertible too. Theorem 2.5. If G is a

292. nite subgroup of GL3 (C) then fB Gg = 1 in cK0(Vark) and the Ekedahl invariants of G are trivial. Proof. Using equation f[P2=G]g = (1 + L + L2) fB Gg and Proposition 2.2, we know that fB GgfP2g = f[P2=H]g. Since fP2g is invertible in cK0(Vark), it is sucient to prove that f[P2=H]g = fP2g. Let U be the open subset of P2 where H acts freely and let C be the complement of U in P2. We denote by C0 and C1 respectively the dimension zero and the dimension one closed subsets of C so that C = C0 t C1. One observes that [C0=H] is the disjoint union of a

293. nite number of quotient stacks [Oi=H] where Oi are the orbits of Pi 2 C0 under the action of H. We note that [Oi=H] = [Pi=StabPi (H)] = B StabPi(H). By Cartan's Lemma, StabPi(H) is a subgroup of GL2 (C) and then, by using Proposition 2.4, f[Oi=H]g = fB StabPi(H)g = fOi=Hg = fg = 1: Therefore f[C0=H]g = fC0=Hg. We observe that f[S=H]g = fS=Hg holds for every

294. nite stable subset S of P2 with the same argument. 8

295. The set C1 is the union of a

296. nite number of lines Li. We denote by I, the subset of C1, made by the intersection points of those lines Li. We also denote by C 1 the complement of I in C1. Let L be a line in C1 and SL = StabL(H). Since H PGL3, then one sees that SL StabL(PGL3) H. We observe that StabL(PGL3) = GL2 nC2 because a class in StabL(PGL3) has the form 2 4 0 @ 1 0 0 GL 2 1 A 3 5 and therefore StabL(PGL3) = GL2 nC2. So one has the group homomor-phism GL2 nC2 ! GL2 sending (g; x) to g. The kernel of such homomor-phism restricted to SL is trivial, because ker(GL2 nC2 ! GL2) = C2 and then ker(SL ! GL2) = SL C2 = 0. Thus, SL GL2 and, using the Proposition 2.4, one gets f[L=SL]g = fL=SLg. We set L0 = L C 1 . Then [L=SL] = [L0=SL] [ [LnL0=SL]. For what we said for 0j the zero dimensional case f[LnL0=SL]g = fLnL0=SLg and so f[L0=SL]g = fL0=SLg. We call Othe orbit of L0 j under H. Since C 1 is the disjoint union of a

297. nite number of orbits O0j , then f[C1=H]g = f[C 1=H]g + f[I=H]g = X j f[O0 j=H]g + f[I=H]g = X j f[L0 j=SLj ]g + f[I=H]g = X j fL0 j=Hg + fI=Hg = fC1=Hg: Summarizing the proven facts, one has f[P2=H]g = f[U=H]g+f[C0=H]g+f[C1=H]g = fU=Hg+fC0=Hg+fC1=Hg = fP2=Hg: Therefore there remains to prove that fP2=Hg = fP2g. For this purpose let X be a resolution of the singularities of P2=H, : X ! P2=H. An unirational surface (over C) is rational and one can construct a birational morphism on P2, 0 : X ! P2. The quotient singularities of P2=H are rational singularities and the exceptional divisor Dy of y 2 Sing (P2=H) is a tree of P1. This implies that Dy = [ny j=1P1, where ny is the number of irreducible components of Dy. Then fDyg = nyfP1g P i;jfg. Since the graph of the resolution is 9

298. a tree, then there are exactly ny 1 intersection points in P i;jfg. Hence fDyg = nyfP1g (ny 1) = nyL + 1. Then, fP2=Hg = fXg X y (fDyg fyg) = fXg X y (nyL + 1 1) = fXg L X y ny = fXg Ln; where n = P y ny is the number of irreducible components in the full excep-tional divisor D = [yDy. Similarly, one gets fP2g = fXg Lm, where m is the number of irreducible components in the full exceptional divisor E of the resolution X 0 ! P2. We shall prove that m = n. Let us consider the following spectral se-quence from the map : X ! P2=H: 2 = Hi Ei;j P2=H;Rj:QX ) Hi+j (X;Q) : Since the map is an isomorphism a part a

299. nite number of points, Rj:QX is de

300. ned over those points (and zero elsewhere). Let y be one of those: (Rj:QX)y = Hi (1(y);Q) and so H0 (1(y);Q) = Qn and for i 0, Hi (1(y);Q) = 0. The spectral sequence degenerates and then we obtain 0 ! Q ! H2 (X;Q) ! Qn ! 0: This implies H2 (X;Q) = Qn+1. Similarly, for 0 : X ! P2, one gets H2 (X;Q) = Qm+1 and, thus, the equality m = n. The diculties to continue such induction up to GL4 (C) arise from the study of the resolution of singularities of P3=H. In general for bigger n, Car-tan's Lemma reduces this question to the study of the quotients Cn1=A for certain A GLn1 and those are not well known. 3 Finite groups with abelian projective reduction As in the previous section G is a subgroup of GLn and H is its reduction in PGLn. If H is abelian and if the singularities of Pn1=H are zero dimensional, then the Ekedahl invariants satisfy a recursive equation. 10

301. Theorem 3.1. Let G be a

302. nite subgroup of GLn (C) and let H be the image of G under the canonical projection GLn (C) ! PGLn(C). If H is abelian and if Pn1=H has only zero dimensional singularities, then for every integer k ek (G) + ek+2 (G) + + ek+2(n1) (G) = fHk (X;Z)g; where X is a smooth and proper resolution of Pn1=H. We

303. rst show a technical lemma. We denote by pX(t) = P i0

304. i(X)ti the virtual Poincare polynomial of a complex algebraic scheme X, where

305. i(X) = dim(Hi (X;Q)) is the i-th Betti number of X. For every smooth projective toric variety Y , the odd degree coecients of pY (t) are zero (Section 5.2 of [6]). In addition, if G is a

306. nite subgroup of GLn as above, then pPn1=H(t) = pPn1(t). Indeed H (Pn1=H;Q) = H (Pn1;Q)H = H (Pn1;Q). Lemma 3.2. Let G, H, Pn1=H and X satisfy the hypothesis of Theorem 3.1. Then: i) fB Gg(1 + L + + Ln1) = fPn1=Hg and, in particular, ek (G) + ek+2 (G) + + ek+2(n1) (G) = Hk (fPn1=Hg) : ii) Every singularity of Pn1=H is a toroidal singularity and fPn1=Hg = fXg X y (fDyg fyg) ; (2) where the sum runs over y 2 Sing (Pn1=H); fDyg is the exceptional divisor of the toric resolution of y with irreducible components decomposi- tion Dy = D1 y[ [Dry ; fDyg = P q1(1)q+1P i1;:::;iq y Diq fDi1 y g. iii) If k is non-zero and even, one has 1 =

307. k(X) X y X q1 (1)q+1 X i1;:::;iq

308. k(Di1 y Diq y ) and, for k = 0, 1 =

309. 0(X) X y X q1 (1)q+1 X i1;:::;iq

310. 0(Di1 : y Diq y ) 1 11

311. iv)

312. odd(X) = 0. Proof. By assumptions Pn1=H has only zero dimensional singularities. Re-garding item i) we observe that f[Pn1=H]g = fPn1=Hg + X j (fB StabPj (H)g fg) where the sum runs over the orbits of points with nontrivial stabilizer in Pn1 and Pj is a point in such an orbit. Every stabilizer group of H is abelian and we know, by Proposition 1.6.2) that fB StabPj (H)g = 1. So f[Pn1=H]g = fPn1=Hg. Using also Proposition 2.2, we obtain the

313. rst part of i). For the second one, we note that applying the cohomological map Hk on the left hand side, one has: Hk fB Gg(1 + + Ln1) = Hk (fB Gg) + + Hk fB GgLn1 = Hk (fB Gg) + + Hk2(n1) (fB Gg) = ek (G) + + ek+2(n1) (G) : Every stabilizer group of H is abelian and so it is for the quotient of Stabx(H) modulo Pseudo (Stabx(H)) in TxX. Then, for Lemma 2.3, each singularity of fPn1=Hg is an isolated simplicial toroidal singularities. One produces a toric resolution with normal crossing toric exceptional divisors (see Section 2.6 of [6]). We mean that calling Dy the exceptional divisor of the resolution of the toroidal singularity y in Pn1=H, Dy = D1 y [ [ Dry y Diq and each intersection Di1 y is a smooth toric varieties. Hence, one has equation (2) and fDyg = P q1(1)q+1P i1;:::;iq y Diq fDi1 y g. Thus, pDy (t) = P q1(1)q+1P i1;:::;iq pD i1 y D iq y (t) and the odd degree coecients of pDy (t) are zero. We want to compute the virtual Poincare polynomial of X. Via formula (2) and using pPn1=H(t) = pPn1(t), pPn1(t) = pX(t) X y (pDy (t) 1): Comparing, degree by degree, the polynomial in the left hand side and in the right hand side, one gets the Betti numbers equalities and item iv). 12

314. Proof of Theorem 3.1. From the

315. rst item of the previous lemma, we know that ek (G) + ek+2 (G) + + ek+2(n1) (G) = Hk (fPn1=Hg) : (3) We shall show that Hk (fPn1=Hg) = fHk (X;Z)g. For this, we study a resolution of the singularities of Pn1=H. Using the previous technical lemma we express P fPn1=Hg in (2) as a sum of smooth and proper varieties and fDyg = q0(1)q+1P i1;:::;iq y Diq fDi1 y g where Dy is the exceptional divisor of the resolution of the singularity y in fPn1=Hg. Moreover Dy = D1 y [ [ Dry , where Djy y Diq and each intersection Di1 y are smooth toric varieties. If k 0 or k 2(n 2), Hk (fDyg fyg) = 0 for dimensional reason and so the recurrence holds. Similarly if k is odd integer between 0 and 2(n2) because the cohomology of a smooth toric variety is torsion free. It remains the case 0 k = 2j 2(n 2). For these values, in the left hand side of (3), there are some negative Ekedahl invariants (so zero), e0 (G) and some positive even Ekedahl invariants e2 (G) + + e2j+2(n1) (G) that are sum (with sign) of classes of

316. nite abelian groups (we use the second part of Theorem 1.4). On the right hand side of (3) the only possible torsion part is ftorHk (X;Z)g, because the cohomologies of a smooth toric variety is torsion free (see Section 5.2 in [6]). Hence, what remains to prove is that the free parts cancel each others: for k6= 0, fZg =

317. k(X)fZg X y X q1 (1)q+1 X i1;:::;iq y Diq y )fZg

318. k(Di1 and fZg =

319. 0(X)fZg X y X q1 (1)q+1 X i1;:::;iq

320. 0(Di1 fZg: y Diq y ) 1 These follow from the Lemma 3.2.iii). 13

321. 4 The discrete Heisenberg group Hp Let p be a prime dierent from two. The p-discrete Heisenberg group Hp is the following subgroup of GL3 (Fp): Hp = 8 : 0 @ 1 a b 0 1 c 0 0 1 1 A : a; b; c 2 Fp 9= ;: If we denote by M(a; b; c) = 0 @ 1 a b 0 1 c 0 0 1 1 A; then we observe that Hp is generated by X = M(1; 0; 0), Y = M(0; 0; 1) and Z = M(0; 1; 0) modulo the relations ZYX = XY, Zp = Xp = Yp = Id, ZX = XZ and ZY = YZ. The center of Hp, ZHp, is generated by Z and we denote by Ap the group quotient Hp=ZHp = Z=pZ Z=pZ. Moreover, Hp is the central extension of Z=pZ by Z=pZ Z=pZ: 1 ! Z=pZ ! Hp ! Z=pZ Z=pZ ! 1; and then, using Lemma 4.9 in [3], one proves that the Bogomolov multiplier B0(Hp) is zero for every prime p. We also remark that the discrete Heisenberg group has p2 +p1 irreducible complex representations: p2 of them are one dimensional and the remaining p 1 are faithful and p-dimensional. Let V be a faithful irreducible p-dimensional complex representation of Hp, Hp ! GLp (C). There is a natural action of Hp on V and it induces an action on Pp1. One so de

322. nes the quotient Pp1=Hp. Since Z belongs to the center, (Z) = e 2i p Idp, for some 0 i p. Hence, the center acts trivially on Pp1 and Pp1=Hp = Pp1=Ap. From Lemma 2.3, we know that if Pp1=Ap has singularities, then they are toroidal. We study these singularities and so we focus on Stabx(Ap). Proposition 4.1. Let x 2 Pp1. If the action of Ap at x is not free, then j Stabx(Ap)j = p. Proof. Let Wx be the one dimensional subvector-space of V corresponding to x. The stabilizer of x is a subgroup of Ap and, by Lagrange's Theorem 14

323. (and using the assumptions), it could have order p or p2. If Stabx(Ap) = Ap, then for every g 2 Ap; gWx = Wx and ApWx = Wx. Then HpWx = Wx. This implies that Wx is an one dimensional irreducible Hp-subrepresentation of V contradicting the fact that Hp acts irreducibly. There are exactly p+1 subgroups of order p in Ap. Let B be one of them. = We de

324. ne B b as a subgroup of Hp such that j B b is a group isomorphism: B b Z=pZ 1(B). We restrict the representation Hp ! GLp (C) to the subgroup b B = Z=pZ. We write V as a direct sum of one dimensional irreducible representations: V = 2Z=pZV, where V = fv 2 V : g v = (g) v; 8g 2 Z=pZg. In other words, b B

325. xes p one dimensional linear subspaces V and so B

326. xes p points P 2 Pp1, with StabP(Ap) = B, that is (Pp1)B = fP0 ; : : : ; Pp1g. Proposition 4.2. If B and B0 are two distinct p-subgroups of Ap, then (Pp1)B (Pp1)B0 = ;. Proof. Trivially, Ap = BB0 and if P 2 (Pp1)B(Pp1)B0 , then StabP (Ap) = Ap, contradicting Proposition 4.1. We observe that Ap=B acts regularly on (Pp1)B. Thus, these points are a unique orbit under the action of Ap=B and this means that they correspond to a unique point yB in Pp1=Ap. Theorem 4.3. The quotient Pp1=Ap has p + 1 simplicial toroidal singular points. = Proof. There are exactly p+1 subgroups, B, of order p in Ap. Each of them corresponds to a point yB in Pp1=Ap. By Proposition 4.2, these points are distinct. Let y 2 Pp1 such that y = yB. We consider the action of Staby(Ap) on the tangent space TyPp1. The pseudo-re ection group Pseudo (Staby(Ap)) is zero, because it is a subgroup of Staby(Ap) Z=pZ and, so, it is either the trivial group or Staby(Ap). The latter is not possible because Staby(Ap) stabi-lizes only the origin of the vector space TyPp1. Thus, Pseudo (Staby(Ap))6= Staby(Ap) in TyPp1 and for Lemma 2.3 these singularities are also toroidal and simplicial. 15

327. We now draw a method to calculate the Ekedahl invariants for Hp: we write fPp1=Apg as a sum of classes of smooth and proper varieties and we use Theorem 3.1. Let Xp f! Pp1=Ap be the resolution of the p+1 toroidal singularities of Pp1=Ap. One has the following geometrical picture: P(V ) U Xp _ f Pp1=Ap ^ [ Up f _ Up where U is the open subset of P(V ) where Ap acts freely; Up = U=Ap and Xp is a smooth and proper resolution of Pp1=Ap. Since Ap is abelian, using Theorem 3.1 one gets ek (G) + ek+2 (G) + + ek+2(p1) (G) = fHk (Xp;Z)g: Because of Theorem 1.4, e0 (Hp) = fZg and e1 (Hp) = e2 (Hp) = 0. Thus, we focus on e3 (Hp). We are going to show that e3 (H5) = e4 (H5) = 0. We set p = 5 because of the hardness to compute tor(H (Xp;Z)), for every p. Claim. tor(H5 (X5;Z)) = 0. Using this, we prove the main results. Theorem 4.4. The Ekedahl invariants of the

328. fth discrete Heisenberg group H5 are trivial. Proof. By using Theorem 3.1 for G = H5, n = 5, k = 2 5 + 5 and X = X5 and also by applying the second part of Theorem 1.4, we have e3 (H5) = ftor H255 (X5; Z) g = ftor H5 (X5;Z) g and then, for duality, e3 (H5) = ftor H5 (X5; :)Z g = ftor H4 (X5;Z) g and this is zero for the claim. Similarly, for k = 2 5 + 6, e4 (H5) = ftor H256 (X5;Z) g = ftor H4 (X5;Z) g: 16

329. Moreover, e3 (H5) and e4 (H5) are the only invariants that could not be zero. Indeed, e5 (H5) = ftor H3 (X5;Z) g = e2 (H5) = 0 and e6 (H5) = ftor H2 (X5; Z) g = e1 (H5) = 0. In addition, ei (H5) = 0 for i 6 for dimensional reason. We observe that the same proof would follow for Hp having enough informa-tion about the vanishing of the torsion in the cohomology of Xp. 4.1 Proof of the claim To obtain the proof of the claim we need to show that Hodd (U5;Z) = 0 and H5 E (X5;Z) = 0 are zero, where E is the union of exceptional divisors of the resolution X5 f! P51=A5. The

330. rst fact is actually true for Up. Theorem 4.5. The cohomology of the smooth open subset Up of P(V )=Ap for k 2p 2 is Hk (Up;Z) = 8 : Z if k = 0; 0 if k is odd; Z (Z=pZ) k 2 +1 if k6= 0 and even: Proof. Since Ap acts freely on U, let us consider the Cartan-Leray spectral sequence (see Section 5 or Theorem 8bis:9 in [9]) relative to the quotient map : U ! Up: 2 = Hi Ei;j ) Hi+j (Up;Z) : Ap;Hj (U;Z) Let Sp = Pp1 n U. One sees that Hi (U;Z) = Hi (P(V ); Z) for i 2p 3, H2p3 (U;Z) = Z[Sp]0 and H2p3 (U;Z) is zero otherwise. Here Z[Sp] is the group freely generated by the p(p + 1) points in Sp and Z[Sp]0 is the kernel of the argumentation map. To read the Ei;j 2 -terms we observe that the cohomology of Ap has a Z-algebra structure: H (Ap;Z) = Z[x1; x2; y] (y2; px1; px2; py) ; where deg(x1) = deg(x2) = 2 and deg(y) = 3. Indeed the Z-algebra structure comes from the Bockstein operator for H (Z=pZ; Fp). (The reader may

331. nd a detailed proof in Appendix A of [7].) For what concerns this proof, we only care about the terms Ei;j 2 with j 2p 3. The dierential di;j 2 is zero if j 2p 3. 17

332. Let h be the

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