# Cycles on curves and Jacobians: a tale of two tautological rings

@inproceedings{Yin2014CyclesOC, title={Cycles on curves and Jacobians: a tale of two tautological rings}, author={Qizheng Yin}, year={2014} }

We connect two notions of tautological ring: one for the moduli space of curves (after Mumford, Faber, etc.), and the other for the Jacobian of a curve (after Beauville, Polishchuk, etc.). The motivic Lefschetz decomposition on the Jacobian side produces relations between tautological classes, leading to results about Faber’s Gorenstein conjecture on the curve side. We also relate certain Gorenstein properties on both sides and verify them for small genera. Further, we raise the question…

## 9 Citations

### Relations in the tautological ring of the universal curve

- MathematicsCommunications in Analysis and Geometry
- 2022

After some background theory we provide a brief summary of what is known about the tautological ring of the moduli space curves. We then formulate a few conjectures about the structure of the…

### Computation of $\lambda$-classes via strata of differentials

- Mathematics
- 2022

. We introduce a new family of tautological relations of the moduli space of stable curves of genus g . These relations are obtained by computing the Poincaré-dual class of empty loci in the Hodge…

### A calculus for the moduli space of curves

- MathematicsAlgebraic Geometry: Salt Lake City 2015
- 2018

This article accompanies my lecture at the 2015 AMS summer institute in algebraic geometry in Salt Lake City. I survey the recent advances in the study of tautological classes on the moduli spaces of…

### An Action of the Polishchuk Differential Operator via Punctured Surfaces

- MathematicsInternational Mathematics Research Notices
- 2020

For a family of Jacobians of smooth pointed curves, there is a notion of tautological algebra. There is an action of ${\mathfrak{s}}l_2$ on this algebra. We define and study a lifting of the…

### Tautological classes with twisted coefficients

- MathematicsAnnales scientifiques de l'École Normale Supérieure
- 2021

Let $M_g$ be the moduli space of smooth genus $g$ curves. We define a notion of Chow groups of $M_g$ with coefficients in a representation of $Sp(2g)$, and we define a subgroup of tautological…

### A CALCULUS FOR THE MODULI SPACE OF CURVES

- Mathematics
- 2018

The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduli…

### Poincaré Duality of Wonderful Compactifications and Tautological Rings

- Mathematics
- 2015

Let $g \geq 2$. Let $M_{g,n}^{rt}$ be the moduli space of $n$-pointed genus $g$ curves with rational tails. Let $C_g^n$ be the $n$-fold fibered power of the universal curve over $M_g$. We prove that…

### Pixton's formula and Abel-Jacobi theory on the Picard stack

- Mathematics
- 2020

Let $A=(a_1,\ldots,a_n)$ be a vector of integers with $d=\sum_{i=1}^n a_i$. By partial resolution of the classical Abel-Jacobi map, we construct a universal twisted double ramification cycle…

### Tautological rings of spaces of pointed genus two curves of compact type

- MathematicsCompositio Mathematica
- 2016

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$ , the moduli space of $n$ -pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$ .…

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