
Open access
Date
2020Type
- Journal Article
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. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000467825Publication status
publishedExternal links
Journal / series
Épijournal de Géométrie AlgébriqueVolume
Pages / Article No.
Publisher
Center for Direct Scientific CommunicationSubject
Chow groups; Moduli spaces of curves; Tautological ringsFunding
182181 - Cohomological field theories, algebraic cycles, and moduli spaces (SNF)
162928 - Cohomology of moduli spaces (SNF)
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