Chow rings of stacks of prestable curves I

TL;DR

This paper investigates the Chow ring structure of the moduli stack of prestable curves, extending intersection formulas and defining tautological classes, laying groundwork for further research in algebraic geometry of stacks.

Contribution

It introduces the notion of tautological classes on the stack of prestable curves and extends intersection and functoriality formulas from stable to prestable cases.

Findings

Defined tautological classes on $rak{M}_{g,n}$

Extended intersection product formulas

Developed theory of proper pushforward for algebraic cycles

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.