Chow rings of stacks of prestable curves I

Abstract

We study the Chow ring of the moduli stack ${\mathfrak{M}}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space ${\bar{\mathcal{M}}}_{g,n}$ of stable curves. This paper provides foundations for the second part of the paper. In the appendix (joint with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of ${\mathfrak{M}}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.

This paper is the first part of the previous version which has been split off due to length. v2. An appendix (joint with Jonathan Skowera) about proper pushforwards for Chow groups of Artin stacks has been added. v3. Major revision on the appendix after referee's report. Results are unchanged. Comments are still very welcome!

References in corpus (1)

• Topological recursion relations and Gromov-Witten invariants in higher genus

Cited by in corpus (4)

• Stable cuspidal curves and the integral Chow ring of ${\overline{\mathcal{M}}}_{2,1}$
• Splitting of Gromov-Witten Invariants with Toric Gluing Strata
• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
• On the tautological rings of stacks of twisted curves