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On the Chow ring of the Hurwitz space of degree three covers of ${\bf P}^{1}$

Anand Patel, Ravi Vakil

Abstract

We determine that the Chow ring (with ${\bf Q}$-coefficients) of the Hurwitz space parametrizing degree three covers of ${\bf P}^{1}$ is tautological. We also compute the rational Picard groups of auxiliary spaces of degree three maps with special marked points.

On the Chow ring of the Hurwitz space of degree three covers of ${\bf P}^{1}$

Abstract

We determine that the Chow ring (with -coefficients) of the Hurwitz space parametrizing degree three covers of is tautological. We also compute the rational Picard groups of auxiliary spaces of degree three maps with special marked points.

Paper Structure

This paper contains 27 sections, 15 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Remark on integral coefficients
  3. Outline of the paper
  4. Notation and conventions
  5. Preliminaries
  6. Trigonal curves as divisors on Hirzebruch surfaces
  7. The Maroni invariant
  8. Expressing an open subset of $\mathcal{H}_{3,g}$ as a quotient
  9. The divisor theory of $\mathcal{H}_{3,g}$
  10. Vistoli's theorem: Chow rings of quotients
  11. Vistoli's theorem
  12. The quotient map $q {\colon} \mathcal{H}^{\dagger}_{3,g} \to \mathcal{H}_{3,g}$ and the class $c_{2}(V)$
  13. The universal ramification point $\mathcal{R}$
  14. Expressing an open substack of $\mathcal{R}$ as a quotient
  15. Jet bundles on the Hirzebruch surface ${\bf F}$
  16. ...and 12 more sections

Key Result

proposition \oldthetheorem

Theorems & Definitions (32)

  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • proposition \oldthetheorem
  • theorem \oldthetheorem: Stankova
  • theorem \oldthetheorem: Vistoli
  • Remark \oldthetheorem
  • lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • ...and 22 more