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Journal Article

#### Zero cycles on the moduli space of curves

##### External Resource

https://doi.org/10.46298/epiga.2020.volume4.5601

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##### Fulltext (public)

Pandharipande-Schmitt_Zero cycles on the moduli space of curves_2020.pdf

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##### Citation

Pandharipande, R., & Schmitt, J. (2020). Zero cycles on the moduli space of curves.* Épijournal de Géométrie Algébrique,* *4*: 12. doi:10.46298/epiga.2020.volume4.5601.

Cite as: https://hdl.handle.net/21.11116/0000-0008-BAB3-C

##### 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.

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.