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Moduli of curves on Enriques surfaces

Ciro Ciliberto, Thomas Dedieu, Concettina Galati, Andreas Leopold Knutsen

Abstract

We compute the number of moduli of all irreducible components of the moduli space of smooth curves on Enriques surfaces. In most cases, the moduli maps to the moduli space of Prym curves are generically injective or dominant. Exceptional behaviour is related to existence of Enriques--Fano threefolds and to curves with nodal Prym-canonical model.

Moduli of curves on Enriques surfaces

Abstract

We compute the number of moduli of all irreducible components of the moduli space of smooth curves on Enriques surfaces. In most cases, the moduli maps to the moduli space of Prym curves are generically injective or dominant. Exceptional behaviour is related to existence of Enriques--Fano threefolds and to curves with nodal Prym-canonical model.

Paper Structure

This paper contains 9 sections, 38 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Moduli spaces of Enriques surfaces
  3. Generalities on moduli maps
  4. Fibers of the moduli maps and Enriques--Fano threefolds
  5. Computing cohomology of twisted tangent bundles
  6. Fiber dimensions of moduli maps
  7. A technical result
  8. The moduli maps on $\mathcal{EC}_{g,2}^{(I)}$ for $g \geqslant 7$
  9. The moduli maps on $\mathcal{EC}_{g,1}$, $\mathcal{EC}_{g,2}^{(II)}$, $\mathcal{EC}_{g,2}^{(II)^+}$ and $\mathcal{EC}_{g,2}^{(II)^-}$

Key Result

Theorem 1

Assume that $\phi \geqslant 3$ (whence $g \geqslant 6$). The map $\chi_{g,\phi}: \mathcal{EC}_{g,\phi} \to \mathcal{R}_g$ is generically injective on any irreducible component of $\mathcal{EC}_{g,\phi}$ not appearing in the list below, for which the dimension of a general fiber is indicated:

Theorems & Definitions (86)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1.1
  • Corollary 1.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 76 more