Point counts and the cohomology of moduli spaces
of curves
JONAS BERGSTRÖM
Doctoral Thesis
Stockholm, Sweden 2006
TRITA-MAT-06-MA-04 ISSN 1401-2278
ISRN KTH/MAT/DA 06/02-SE ISBN 91-7178-447-0
KTH Matematik SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 29 september 2006 klockan 14.00 i Nya kollegiesalen, F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm.
Jonas Bergström, september 2006 Tryck: Universitetsservice US AB
Abstract. In this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information.
Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of M4.
In Paper II we consider the moduli space Hg,nof n-pointed smooth
hyper-elliptic curves of genus g. We find that there are recursion formulas in the genus that the numbers of points of Hg,nfulfill. Thus, if we can make Sn-equivariant
counts of Hg,nfor low genus, then we can do this for every genus. Information
about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to M2,n for n up to seven, and give us the Sn
-equivariant Hodge structure of their cohomology. Moreover, we find that the Sn-equivariant counts of Hg,ndepend upon whether the characteristic is even
or odd, where the first instance of this dependence is for six-pointed curves of genus three.
In Paper III we consider the moduli space Qnof smooth n-pointed
non-hyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make Sn-equivariant counts of the numbers of
points of Qn for n up to seven. We also count pointed plane cubics. This
gives us Sn-equivariant counts of the moduli space M1,nfor n up to ten. We
can then determine the Sn-equivariant Hodge structure of the cohomology of
Contents
Introduction 1
1. Cohomology of moduli spaces of pointed curves 1
2. Counting curves over finite fields 2
3. Outline of results 3
4. Acknowledgements 5
Bibliography 7
Paper I. The rational cohomology of M4
1. Introduction and results 1
2. The stratification of MG,N 5
3. From counting to cohomology 9
4. Counting points over finite fields 13
5. Rational cohomology of geometric quotients 19
6. Vassiliev-Gorinov’s method 23
References 30
Paper II. Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves
1. Introduction 1
2. Equivariant counts 2
3. Representatives of hyperelliptic curves in odd characteristic 4 4. Recursive equations for ugin odd characteristic 8
5. Linear recursions for aR1
N1. . . a RM
NM|g 12
6. Computing u0 13
7. Results for weight up to seven in odd characteristic 14 8. Representatives of hyperelliptic curves in even characteristic 16 9. Recursive equations for ugin even characteristic 18
10. Results for weight up to seven in even characteristic 23 11. Euler characteristics in the case of genus two 24
12. Appendix: Introducing bi, ci and ri 27
References 30
Paper III. Cohomology of moduli spaces of curves of genus three via point counts
1. Introduction 1
vi CONTENTS
2. The moduli space of stable curves of genus three 2
3. Preliminaries 4
4. Quartic curves 5
5. The sieve principle 7
6. Tools 9
7. The library of singular degree four curves 11
8. Infinitely many singularities 13
9. Six singularities 15 10. Five singularities 16 11. Four singularities 17 12. Three singularities 20 13. Two singularities 29 14. Linear subspaces 40
15. Cubic curves and elliptic curves 46
16. Trace of Frobenius on some local systems on Q 50
Introduction
In this thesis we find information on the cohomology of certain moduli spaces of pointed curves by counting their numbers of points defined over finite fields. We will first introduce the most important notions of the thesis and then present our results.
1. Cohomology of moduli spaces of pointed curves
Let us consider the curves, that is the one dimensional algebraic varieties de-fined over an algebraically closed field, that are proper, irreducible and smooth. As a first step in classifying these curves we distinguish them according to a dis-crete invariant called the genus, which is a non-negative integer. The genus can be defined as the dimension of the space of regular differentials of the curve. It can also be visualized if we are considering a curve defined over the complex numbers, because then the curve is topologically a compact Riemann surface with as many holes as its genus.
For reasons that will become clearer below, what we would actually like to classify is pointed curves. An pointed curve is a curve C together with an n-tuple of distinct marked points (p1, . . . , pn). With a morphism from (C, p1, . . . , pn)
to (D, q1, . . . , qn) we then mean a morphism from C to D that sends pi to qi for all
i. If 2g − 2 + n > 0 then the automorphism group of an n-pointed curve of genus g is finite. From now on, let g and n be non-negative integers fulfilling this condition. The starting point for this thesis is that there exists a moduli space Mg,n that
classifies the curves of genus g together with an n-tuple of marked points, see [8]. That is, the points of Mg,ncorrespond precisely to the isomorphism classes of curves
of genus g together with n marked points and moreover, from a family of curves of genus g together with n marked points over a base S we get an induced morphism from S to Mg,n. This space is unfortunately not in itself an algebraic variety but
something eerier, namely a stack. In fact it is a smooth Deligne-Mumford stack of dimension 3g − 3 + n over the integers. To this stack we can associate its so called coarse moduli space ˜Mg,nwhich is indeed an algebraic variety. But this algebraic
variety is in general not smooth.
The moduli space Mg,n (or for that matter ˜Mg,n) is not proper, or in the
language of topology, not compact. Properness is a very desirable property and to attain it we will enlargen the category of curves we classify to include the stable n-pointed curves. A stable n-pointed curve is a reduced, connected curve with at
2 INTRODUCTION
worst nodal singularities together with n distinct marked smooth points and such that for each smooth rational component, the number of marked points on this component plus the number of times it meets the rest of the curve, is at least three. The moduli space Mg,n of stable n-pointed curves exists and is a proper
and smooth Deligne-Mumford stack that contains Mg,nas a Zariski open part, see
[4] and [6].
Of importance to us, is that the complement ∂Mg,nof Mg,n inside Mg,n can
be stratified into pieces that are products, of moduli spaces M˜g,˜n for ˜g ≤ g and
˜
n ≤ n + g − ˜g, divided by a finite group. Hence, even if we study the moduli space Mgof non-pointed stable curves, we find that the moduli spaces of pointed smooth
curves naturally appear.
What we would like to understand is the cohomology of these moduli spaces. For an overview of some of the results on this matter see for instance the intro-duction of [1] and also the introintro-ductions of Papers I, II and III in this thesis. A nonsingular algebraic variety over the complex numbers can also be considered as a complex manifold, the cohomology is then a very important invariant of the mani-fold together with its complex topology. The genus of a smooth curve is for instance equal to half the dimension of the first cohomology group. The cohomology groups of a complex space also (often) come with an important additional structure called the Hodge structure. We note here in passing that the rational cohomology of a stack and the rational cohomology of its coarse moduli space are the same.
The method we will employ to find cohomological information will be to count points over finite fields.
2. Counting curves over finite fields
For each prime power q there is a finite field k with q elements. The group of automorphisms of the algebraic closure ¯k of k contains the q:th power map, which is commonly called the Frobenius map. We find that the fixed points of the Frobenius map are precisely the elements of the finite field k.
One could wonder how many ¯k-isomorphism classes there are of smooth n-pointed curves of genus g that are defined over the finite field k. This is a finite number, and it is equal to the number of fixed points of the Frobenius map acting on the coarse moduli space of Mg,n extended to ¯k. Even though the world over
finite fields is very different from the world over the complex numbers, these counts actually give information on the cohomology of Mg,n over the complex numbers.
The connection was established via the construction of ´etale cohomology by Grothendieck. If we for instance are considering a smooth algebraic variety over the complex numbers then the ´etale cohomology is equal to the ordinary complex cohomology. Moreover, the ´etale cohomology is endowed with a Lefschetz fixed point theorem. This theorem tells us, among other things, that the number of fixed points of Frobenius acting on an algebraic variety defined over k and extended to ¯
k, is equal to the trace of Frobenius on the Euler characteristic of the compactly supported ´etale cohomology with coefficients in the field of `-adic numbers. If we
3. OUTLINE OF RESULTS 3
are considering a smooth and proper algebraic variety then we also know that the eigenvalues of Frobenius acting on the i:th cohomology group will have absolute value equal to q raised to the power i/2, see [3]. Even though this was formulated for algebraic varieties, these results can to a large extent be generalized to Deligne-Mumford stacks, see [2] for the Lefschetz fixed point theorem, and also [7].
In this thesis we will repeatedly use a variant of the following theorem by van den Bogaart-Edixhoven (which is Corollary 5.3 of [10]) to get cohomological information on Mg,n and Mg,n. Say that we are considering a stack which is “as
nice as” Mg,nand such that the count of points is a polynomial when considered as
a function of the number of elements of the finite field. Then the Hodge structure of each cohomology group of this stack can be determined. In fact, the cohomology will all be of Tate type, and the dimensions of the cohomology groups can be read off from the coefficients of the polynomial.
In order to count the points over finite fields of Mg,n we will use the
stratifica-tion menstratifica-tioned above. That is, formulate the count in terms of informastratifica-tion about moduli spaces of pointed smooth curves. It suffices, by a result of Getzler-Kapranov that is stated below, to take the action of the symmetric group into account. It acts on these moduli spaces by permuting the marked points on the curves. We will therefore make our counts of points equivariant with respect to the symmetric group.
3. Outline of results
Paper I is joint work with Orsola Tommasi. There we first present a formula from the article [5] by Getzler-Kapranov for expressing the Euler characteristic of Mg,n in terms of Sn˜-equivariant Euler characteristics of M˜g,˜n for ˜g ≤ g and
˜
n ≤ n + g − ˜g. Since the trace of Frobenius on the Euler characteristic of compactly supported ´etale cohomology is equal to the number of points over the corresponding finite field, this formula can also be used to count the points of moduli spaces of stable pointed curves in terms of moduli spaces of pointed smooth curves. We then prove the following variant of the theorem of van den Bogaart-Edixhoven discussed above. Say that we have made an Sn-equivariant count of Mg,n for
every finite field. If these counts of points, when considered as functions of the number of elements of the finite field, are polynomials, then we can conclude the Sn-equivariant Hodge structure of the cohomology groups of Mg,n. Recall from above that the cohomology will all be of Tate type. Moreover, if the counts of points of the individual M˜g,˜n are polynomial, then we can also determine the Sn˜
-equivariant Hodge Euler characteristic of these spaces.
Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of comple-ments of discriminants in complex vector spaces. Results obtained on genus four curves by Orsola Tommasi using this method, see [9], together with results obtained in Paper II and Paper III of this thesis, enable us to determine the Hodge structure of the cohomology of M4.
4 INTRODUCTION
In Paper II we consider the moduli space Hg,nof n-pointed smooth hyperelliptic
curves of genus g, which is a closed subset of Mg,n. The smooth hyperelliptic curves
are characterised by their (unique) degree two morphism to the projective line. In odd characteristic, we can use this morphism to describe the curves of genus g that are defined over k as degree two extensions of the form y2
= f (x), where x is a coordinate on the projective line and f is a square-free polynomial of degree 2g + 1 or 2g + 2 with coefficients in k. The Sn-equivariant count of points over k
of Hg,n can then be formulated in terms of these square-free polynomials. In even
characteristic we later find a similar but slightly more complicated description. The main result of this paper is that by using this description of hyperelliptic curves, we can determine recursion formulas in the genus that the Sn-equivariant
counts of points of Hg,nfulfill, see Theorem 5.1 for odd characteristic and Theorem
9.14 for even characteristic. This has the consequence that if we can make Sn
-equivariant counts of Hg,n for low genus, then we can do this for every genus.
The recursions for the numbers of points of Hg,n were based upon other
re-cursions for numbers that contain more information, see the Appendix together with Theorem 4.9 for odd characteristic, and Theorem 9.11 for even characteristic. Namely, they are Sn1 × Sn2-equivariant counts of the number of ¯k-isomorphism
classes of hyperelliptic curves over k together with an n1-tuple of distinct marked
points and an n2-tuple of distinct ramification points.
Information that we have about curves of genus zero and one is then found to be sufficient to make Sn-equivariant counts of Hg,n for all genera and with up to
seven points. We find that the Sn-equivariant counts of Hg,ndepend upon whether
the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three.
All curves of genus two are hyperelliptic and hence we get Sn-equivariant counts
of M2,nfor n up to seven, using the techniques described in Paper I and counts of
pointed curves of genus zero and one. Since these counts are all polynomial we can determine the Sn-equivariant Hodge structure of the cohomology of M2,nfor n up
to seven.
In Paper III we consider the moduli space Qn of smooth n-pointed
nonhyper-elliptic curves of genus three. Using the canonical linear system we can embed such curves into the plane as n-pointed smooth quartic curves.
Identify the space of plane degree four curves defined over k with P14
(k). For each n-tuple of points P we can consider the linear subspace of P14
(k) of degree four curves that contain P . What we would like to find is how many nonsingular curves there are in this subspace. For each point q in the plane, remove the linear subspace of all curves that contain P and that are singular at q. Each curve has then been removed the same number of times as its number of singularities over k. We therefore need to find enough information on the curves with two singularities or more to be able to amend for the incorrect number of times they have been removed.
Using the method sketched above, we make Sn-equivariant counts of the
4. ACKNOWLEDGEMENTS 5
counts of pointed nonsingular plane cubics, which gives us Sn-equivariant counts of
the moduli space M1,n for n up to ten. Together with the results on hyperelliptic
curves of genus two and three made in Paper II and using the techniques of Paper I we can determine the Sn-equivariant Hodge structure of the cohomology of M3,n
for n up to five.
4. Acknowledgements
I am very grateful to my advisor Carel Faber for all help and support. I would like to thank Orsola Tommasi for a very nice collaboration. Thanks also go out to my fellow Ph.D students and roommates Lars Halvard Halle and Jonas S¨oderberg.
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