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Chow classes of divisors on stacks of pointed hyperelliptic curves

Dan Edidin, Zhengning Hu

Abstract

We calculate the classes of the universal hyperelliptic Weierstrass divisor $\bar{\mathcal H}_{g,w}$ and the universal $g^1_2$ divisor $\bar{\mathcal H}_{g,g^1_2}$. Our results are expressed in terms of a basis for $Cl(\bar{\mathcal H}_{g,1})$ and $Cl(\bar{\mathcal H}_{g,2})$ computed by Scavia.

Chow classes of divisors on stacks of pointed hyperelliptic curves

Abstract

We calculate the classes of the universal hyperelliptic Weierstrass divisor and the universal divisor . Our results are expressed in terms of a basis for and computed by Scavia.

Paper Structure

This paper contains 24 sections, 17 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stacks of hyperelliptic curves
  3. Stable hyperelliptic curves as admissible covers
  4. The Weierstrass divisor in $\overline{{\mathcal{H}}}_{g,1}$
  5. The $g^1_2$ divisor in $\overline{{\mathcal{H}}}_{g,2}$
  6. Divisor class groups of stacks of hyperelliptic curves
  7. The divisor class group of $\overline{{\mathcal{H}}}_g$
  8. The divisor class groups of $\overline{{\mathcal{H}}}_{g,1}$ and $\overline{{\mathcal{H}}}_{g,2}$
  9. Intersection of $\overline{{\mathcal{H}}}_{g,w}$ with the boundary of $\overline{{\mathcal{H}}}_{g,1}$
  10. Intersection of $\overline{{\mathcal{H}}}_{g,g^1_2}$ with the boundary of $\overline{{\mathcal{H}}}_{g,2}$
  11. Methods of constructing test curves
  12. Test curves with trivial moduli
  13. A special pencil of pointed hyperelliptic curves with non-trivial moduli
  14. A general construction of pencils of pointed hyperelliptic curves with at worst nodes as singularities
  15. The test family $F_{2i+1}$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Figures (2)

  • Figure 1: Two non--trivial automorphisms
  • Figure 2: The branch locus

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Example 2.2: Andrea Di Lorenzo
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 3.1
  • ...and 25 more