# Zero cycles on the moduli space of curves

## TL;DR

This paper investigates when points on moduli spaces of curves are tautological 0-cycles, providing criteria for rational and K3 surfaces, and exploring connections with Gromov-Witten theory and tautological cycles.

## Contribution

It establishes new conditions under which moduli points of curves on rational and K3 surfaces are tautological 0-cycles, linking these to Gromov-Witten invariants and Beauville-Voisin points.

## Key findings

- Moduli points on rational surfaces are tautological if markings do not exceed Gromov-Witten virtual dimension.
- On K3 surfaces, moduli points are tautological if markings do not exceed genus and are Beauville-Voisin points.
- Several results relate tautological 0-cycles on moduli spaces to geometric properties of the underlying surfaces.

## Abstract

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed.