{"docId":6758,"paperId":5601,"url":"https:\/\/epiga.episciences.org\/5601","doi":"10.46298\/epiga.2020.volume4.5601","journalName":"\u00c9pijournal de G\u00e9om\u00e9trie Alg\u00e9brique","issn":"","eissn":"2491-6765","volume":[{"vid":392,"name":"Volume 4"}],"section":[],"repositoryName":"arXiv","repositoryIdentifier":"1905.00769","repositoryVersion":3,"repositoryLink":"https:\/\/arxiv.org\/abs\/1905.00769v3","dateSubmitted":"2019-06-26 11:19:40","dateAccepted":"2020-09-03 20:58:24","datePublished":"2020-09-03 21:00:12","titles":["Zero cycles on the moduli space of curves"],"authors":["Pandharipande, Rahul","Schmitt, Johannes"],"abstracts":["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"],"keywords":["Mathematics - Algebraic Geometry","14C25, 14H10"]}