Pandharipande, Rahul and Schmitt, Johannes - Zero cycles on the moduli space of curves

epiga:5601 - Épijournal de Géométrie Algébrique, September 3, 2020, Volume 4
Zero cycles on the moduli space of curves

Authors: Pandharipande, Rahul and Schmitt, Johannes

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.


Volume: Volume 4
Published on: September 3, 2020
Submitted on: June 26, 2019
Keywords: Mathematics - Algebraic Geometry,14C25, 14H10


Share

Consultation statistics

This page has been seen 42 times.
This article's PDF has been downloaded 29 times.