Tautological classes on the moduli space of hyperelliptic curves with rational tails
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 of smooth curves of genus . 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 and [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 over [41] and Pixton's conjecture [30] for the Deligne–Mumford compactification of . These conjectures are wide open in full generality. The tautological rings of for 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 for is due to Yin. From his conjecture in [41] the tautological ring of 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 . 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 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 . 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 over . Here we consider the space 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 over the space . The reduced fiber of over a moduli point associated with a smooth hyperelliptic curve X is the Fulton–MacPherson compactification of the configuration space of n points on X introduced in [7]. The tautological ring of the fiber 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 is naturally isomorphic to the tautological ring of the fiber .
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 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 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 -coefficients.
Section snippets
Tautological classes on the space of hyperelliptic curves
Let be the space of stable n-pointed hyperelliptic curves of genus with rational tails for a natural number n. Recall that 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 , for , 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.
Products of the universal curve over
In this section we give a description of the tautological rings for products of the universal hyperelliptic curve over . 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 be the universal hyperelliptic curve of genus g. The cycle vanishes.
Proof Let be the universal curve over the space together with the section
The Fulton–MacPherson compactification
In previous sections we described tautological relations on the moduli space for all g and n and we proved that the restriction map 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 for all g and n, and prove that the restriction map to the Fulton–MacPherson compactification is an isomorphism for any hyperelliptic curve X. After this paper
The tautological ring of
We have studied the tautological algebra of the Fulton–MacPherson space 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 . In particular, this would imply that the tautological ring of the moduli space has the same description as 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 . 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 map The image of is called the tautological cohomology of and is denoted by . From Theorem 0.1 we get the following isomorphism: This shows that
Final remarks
We have found relations among tautological classes on the space . 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 , 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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