Tautological classes on the moduli space of hyperelliptic curves with rational tails

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Abstract

We study tautological classes on the moduli space of stable n-pointed hyperelliptic curves of genus g with rational tails. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. Our result gives a complete description of tautological relations. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology.

Introduction

The study of tautological classes on spaces of curves was initiated by Mumford in his seminal article [23]. Tautological classes are natural algebraic cycles reflecting the nature of the generic object parameterized by the moduli space. The set of generators consists of an explicit collection of cycles. Tautological groups are finite dimensional vector spaces. This distinguishes remarkably the tautological ring from the out of reach space of all algebraic cycles. A basic question about tautological algebras is to give a meaningful class of relations among tautological classes. In [5] Faber introduced a method to produce tautological relations on the moduli space Mg of smooth curves of genus g>2. He conjectured that his method produces all tautological relations and the resulting algebra is Gorenstein. Analogous conjectures were formulated for spaces of pointed curves by Faber and Pandharipande. See [6] for a survey. There are counterexamples for the Gorenstein conjectures on M2,20 and M2,8ct [27], [29]. However, for the spaces of curves with rational tails they are still open. More recent conjectures are Yin's conjecture for the universal curve Cg over Mg [41] and Pixton's conjecture [30] for the Deligne–Mumford compactification Mg,n of Mg,n. These conjectures are wide open in full generality. The tautological rings of Mg for g<24 were computed by Faber. In genus 24 all known methods fail to give a Gorenstein algebra. The analysis of the tautological ring of the universal curve Cg for g<20 is due to Yin. From his conjecture in [41] the tautological ring of C20 should not be Gorenstein. See also the tables by Pixton [31] for more cases that his conjecture contradicts the Gorenstein conjecture. Tautological rings of pointed spaces have been computed in genus at most two [17], [27], [25], [38], [36], [37]. See also [39] for computations in higher genus. In our previous work [37] we studied the tautological ring of the moduli space M2,nrt. Our analysis was based on a series of tautological relations discovered by Faber, Getzler, Belorousski and Pandharipande [4], [8], [2]. A new relation of degree 3 involving 6 points was also found. It was shown that these relations lead to a complete description of tautological relations on M2,nrt for every n. We were not able to give a uniform proof of these relations and in each case the proofs were involved.

Pixton's relations [30] are conjectured to produce all relations on Mg,n. However there is no general method to produce tautological relations on the space of special curves. The goal of this article is to analyse the tautological ring in the simplest case of hyperelliptic curves. The study of more general cases is the subject of upcoming articles. Our study is based on the thesis of Yin [41] who studied the connection between tautological classes on moduli of curves and the universal Jacobian. Tautological classes on both parts were studied extensively. Nevertheless their connection was not understood before his work. In his analysis Yin studied tautological classes on the universal curve Cg over Mg. Here we consider the space Hg,nrt which classifies stable n-pointed hyperelliptic curves of genus g with rational tails. We show that all tautological relations can be easily obtained from relations on the universal Jacobian Jg over the space Hg. The reduced fiber of Hg,nrt over a moduli point [X] associated with a smooth hyperelliptic curve X is the Fulton–MacPherson compactification X[n] of the configuration space of n points on X introduced in [7]. The tautological ring of the fiber X[n] is defined naturally by restriction. We will show the following statement:

Theorem 0.1

Let X be a fixed hyperelliptic curve of genus g. The tautological ring of the moduli space Hg,nrt is naturally isomorphic to the tautological ring of the fiber X[n].

From Theorem 0.1 we obtain a complete presentation of the tautological ring in terms of generators and relations. It also follows that the tautological ring has the Gorenstein property. This means that the tautological ring has a form of Poincaré duality. Our proof gives a uniform proof of the previous result in genus two as well as for hyperelluptic curves in all genera. Everything mentioned above concerns tautological classes in Chow. The Gorenstein property of R(Hg,nrt) implies the same results in cohomology. Using a result of Petersen and Tommasi, which was our motivation for this project, we prove the following:

Corollary 0.2

The cycle class map induces an isomorphism between the tautological ring of the moduli space Hg,nrt in Chow and monodromy invariant classes in cohomology.

Conventions 0.3

We work over an algebraically closed field of characteristic different from 2, 3. We consider algebraic cycles modulo rational equivalence. Chow rings and cohomology rings are taken with Q-coefficients.

Section snippets

Tautological classes on the space of hyperelliptic curves

Let Hg,nrt be the space of stable n-pointed hyperelliptic curves of genus g2 with rational tails for a natural number n. Recall that (C;x1,,xn) is said to be an n-pointed stable curve of genus g if it satisfies the following conditions: the curve C is a nodal curve of arithmetic genus g and the markings xi, for 1in, belong to its smooth locus. Nodes and the markings on each irreducible component of C are called special points. The stability condition implies that all rational components

Tautological classes on the universal Jacobian

The tautological ring of a fixed Jacobian variety under algebraic equivalence is defined and studied by Beauville [1]. We consider the class of a curve inside its Jacobian and apply to it all natural operators induced from the group structure on the Jacobian and the intersection product in the Chow ring. The following result is due to Beauville [1]:

Theorem 2.1

The tautological ring of a Jacobian variety is finitely generated. Furthermore, it is stable under the Fourier–Mukai transform.

In fact, if one

Products of the universal curve over Hg

In this section we give a description of the tautological rings for products Cn of the universal hyperelliptic curve π:CHg over Hg. We find three basic relations among tautological classes. The first two vanishings are probably known to experts. Since they have not appeared in the literature their proofs are given below.

Proposition 3.1

Let π:CHg be the universal hyperelliptic curve of genus g. The cycle K1K2(2g2)K1d1,2 vanishes.

Proof

Let π:CWg be the universal curve over the space Wg together with the section s:

The Fulton–MacPherson compactification

In previous sections we described tautological relations on the moduli space Cn for all g and n and we proved that the restriction map R(Cn)R(Xn) is an isomorphism for any hyperelliptic curve X. In this and the next section we will use these results to determine the structure of the tautological rings R(Hg,nrt) for all g and n, and prove that the restriction map R(Hg,nrt)R(X[n]) to the Fulton–MacPherson compactification is an isomorphism for any hyperelliptic curve X. After this paper

The tautological ring of Hg,nrt

We have studied the tautological algebra of the Fulton–MacPherson space X[n] for a fixed hyperelliptic curve X and a natural number n. The construction of the Fulton–MacPherson compactification works in the relative settings as well. That makes it possible to prove analogous results for the moduli space Hg,nrt. In particular, this would imply that the tautological ring of the moduli space Hg,nrt has the same description as R(X[n]) studied in the previous section. We refer the reader to [26]

Tautological cohomology via monodromy

We have seen a complete description of relations among tautological classes in the Chow ring of Hg,nrt. From the Gorenstein property of the tautological ring there will be no more relations between tautological cycles in cohomology. More precisely, consider the cycle class mapcl:A(Hg,nrt)H(Hg,nrt). The image of R(Hg,nrt) is called the tautological cohomology of Hg,nrt and is denoted by RH(Hg,nrt). From Theorem 0.1 we get the following isomorphism:cl:R(Hg,nrt)RH(Hg,nrt). This shows that

Final remarks

We have found relations among tautological classes on the space Hg,nrt. A natural question is whether these relations extend to larger compactifications. As we have seen in Corollary 5.2 to answer this question it suffices to extend the vanishing of the Faber–Pandharipande and Gross–Schoen cycles. In [30] Pixton proposed a class of conjectural tautological relations on the Deligne–Mumford space Mg,n, which are now known to be true relations [16], [33]. He also conjectures that his method gives

Acknowledgements

I would like to thank Gabriel C. Drummond-Cole, Carel Faber, Gerard van der Geer, Richard Hain, Robin de Jong, Felix Janda, Nicola Pagani, Aaron Pixton and Orsola Tommasi for the valuable discussions and their comments. Special thanks are due to Qizheng Yin for useful discussions and corrections. This research was started during my stay at the Max-Planck-Institut für Mathematik and was completed at the university of Amsterdam. Thanks to Sergey Shadrin for his interest in this project and his

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