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Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces

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

Abstract

We give an explicit description of the irreducible components of the moduli spaces of polarized Enriques surfaces in terms of decompositions of the polarization as an effective sum of isotropic classes. We prove that infinitely many of these components are unirational (resp. uniruled). In particular, this applies to components of arbitrarily large genus $g$ and $φ$-invariant of the polarization.

Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces

Abstract

We give an explicit description of the irreducible components of the moduli spaces of polarized Enriques surfaces in terms of decompositions of the polarization as an effective sum of isotropic classes. We prove that infinitely many of these components are unirational (resp. uniruled). In particular, this applies to components of arbitrarily large genus and -invariant of the polarization.

Paper Structure

This paper contains 5 sections, 26 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Background results on moduli spaces
  3. Background results on line bundles on Enriques surfaces
  4. Simple, isotropic decompositions
  5. Irreducibility, unirationality and uniruledness of moduli spaces

Key Result

Theorem 1.1

The locus of pairs $(S,H) \in {\mathcal{E}}_{g,\phi}$ admitting the same simple decomposition type of length $n\leqslant 4$ is an irreducible, unirational component of ${\mathcal{E}}_{g,\phi}$. The locus of pairs $(S,H) \in {\mathcal{E}}_{g,\phi}$ admitting the same simple decomposition type of leng

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Proposition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • ...and 48 more