10.46298/epiga.2020.volume4.5601
Pandharipande, Rahul
Rahul
Pandharipande
Schmitt, Johannes
Johannes
Schmitt
Zero cycles on the moduli space of curves
episciences.org
2020
Mathematics - Algebraic Geometry
14C25, 14H10
contact@episciences.org
episciences.org
2019-06-26T11:19:40+02:00
2021-02-08T12:04:29+01:00
2020-09-03
eng
Journal article
https://epiga.episciences.org/5601
arXiv:1905.00769
2491-6765
PDF
1
Épijournal de Géométrie Algébrique ; Volume 4 ; 2491-6765
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.
Comment: Published version