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Stable cuspidal curves and the integral Chow ring of $\overline{\mathscr{M}}_{2,1}$

Andrea Di Lorenzo, Michele Pernice, Angelo Vistoli

Abstract

In this paper we introduce the moduli stack $\widetilde{\mathscr{M}}_{g,n}$ of $n$-marked stable at most cuspidal curves of genus $g$ and we use it to determine the integral Chow ring of $\overline{\mathscr{M}}_{2,1}$. Along the way, we also determine the integral Chow ring of $\overline{\mathscr{M}}_{1,2}$.

Stable cuspidal curves and the integral Chow ring of $\overline{\mathscr{M}}_{2,1}$

Abstract

In this paper we introduce the moduli stack of -marked stable at most cuspidal curves of genus and we use it to determine the integral Chow ring of . Along the way, we also determine the integral Chow ring of .

Paper Structure

This paper contains 21 sections, 52 theorems, 178 equations, 1 figure.

Table of Contents

  1. The stack of stable $A_2$-curves of fixed genus
  2. The locus of cusps in the universal curve
  3. The cuspidal locus in an $A_{2}$-curve
  4. The pinching construction
  5. The description of the cuspidal locus
  6. The Chow ring of $\overline{\mathscr{M}}_{1,2}$
  7. The Chow ring of $\widetilde{\mathscr C}_{1,1}$
  8. The Chow ring of $\overline{\mathscr{M}}_{1,2}$
  9. The Chow ring of $\widetilde{\mathscr C}_2$
  10. The Chow ring of $\widetilde{\mathscr C}_2\smallsetminus\widetilde{\Theta}_1$
  11. The Chow ring of $\widetilde{\mathscr C}_2\smallsetminus\widetilde{\Theta}_2$
  12. The Chow ring of $\widetilde{\mathscr C}_2$
  13. The Chow ring of $\overline{\mathscr{M}}_{2,1}$, abstract characterization
  14. Relations from the locus of curves with two cusps
  15. Relations from the locus of curves with one cusp
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose that the ground field has characteristic $\neq 2,3$. Then we have where and the relations are

Figures (1)

  • Figure 1: The pinching construction

Theorems & Definitions (97)

  • Theorem : \ref{['thm:chow Mbar21']}
  • Theorem : \ref{['thm:chow Mbar12']}
  • Definition 1.1
  • Proposition 1.2
  • proof
  • Theorem 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • ...and 87 more