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Nonemptiness of Severi varieties on Enriques surfaces

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

Abstract

Let $(S,L)$ be a general polarized Enriques surface, with $L$ not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible $δ$-nodal curves in the linear system $|L|$, with $0\leq δ\leq p_a(L)-1$. This solves a classical open problem and gives a positive answer to a recent conjecture of Pandharipande--Schmitt, under the additional condition of non-2-divisibility.

Nonemptiness of Severi varieties on Enriques surfaces

Abstract

Let be a general polarized Enriques surface, with not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible -nodal curves in the linear system , with . This solves a classical open problem and gives a positive answer to a recent conjecture of Pandharipande--Schmitt, under the additional condition of non-2-divisibility.

Paper Structure

This paper contains 22 sections, 15 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Flat limits of Enriques surfaces
  3. Logarithmic Severi varieties
  4. Families of blown-up surfaces
  5. Logarithmic Severi varieties on blow--ups of the symmetric square of an elliptic curve
  6. Logarithmic Severi varieties on blown up planes
  7. Deforming to rigid elliptic curves
  8. Isotropic $10$-sequences and simple isotropic decompositions
  9. Isotropic $10$-sequences on members of ${\mathcal{D}}$
  10. Proof of Theorem \ref{['thm:main']}
  11. The case $a_1=a_2=1$ and $a_i=0$ otherwise
  12. The case $a_0=a_9$ and $a_i=0$ otherwise
  13. Subcase $a_0=a_9=1$
  14. Subcase $a_0=a_9 \geq 3$
  15. The cases $a_7=a_9=a_{10}=a_0=0$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $(S,L)$ be a general element of any irreducible component of ${\mathcal{E}}_{g} \setminus {\mathcal{E}}_g[2]$. Then $V_{|L|,\delta}(S)$ is nonempty and has a regular component, of dimension $g-1-\delta$, for all $0 \leq \delta<g$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.6
  • ...and 47 more