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

Abstract

We study tautological classes on the moduli space of stable n-pointed hyperelliptic curves of genus g with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. 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. The connection with recent conjectures by Pixton is also discussed. CONTENTS TAUTOLOGICAL CLASSES ON THE MODULI SPACE H rt g,n 3 Everything mentioned above concerns tautological classes in Chow. The Gorenstein property of R * (H rt g,n ) implies the same results in cohomology. This shows that there is no difference between Chow and cohomology as long as we restrict to tautological classes. 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 H rt g,n in Chow and monodromy invariant classes in cohomology.At the end we discuss the connection between the relations on the space H rt g,n and Pixton's relations on M g,n . Conventions 0.3. We consider algebraic cycles modulo rational equivalence. All Chow groups are taken with Q-coefficients.

Algebraic cycles and intersections of three quadrics

“…This is inspired by the analogous result for cubic hypersurfaces [14, section 2 342,43] (cf. Remark 5·5 below) and for K3 surfaces [50].…”

“…Proof. This is inspired by the analogous result for cubic hypersurfaces [14, section 2.3], which in turn is inspired by analogous results for hyperelliptic curves [42,43] (cf. Remark 5•5 below) and for K3 surfaces [50].…”

“…). Tavakol has proven [43, corollary 6·4] that if C is a hyperelliptic curve, the cycle class map induces injections On the other hand, there are many (non hyperelliptic) curves for which the tautological rings or do not inject into cohomology (this is related to the non-vanishing of the Faber–Pandharipande cycle of Remark 3·16; it is also related to the non-vanishing of the Ceresa cycle, cf. [43, remark 4·2] and [15, example 2·3 and remark 2·4]).…”

“…This is similar to existing results for hyperelliptic curves and for K3 surfaces (cf. [42,43,50] and Remark 5·5 below).…”

Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

“…The following proposition, essentially due to Tavakol [64], relates for a given curve the existence of an MCK decomposition to the existence of enough relations in the tautological ring.…”