Nonemptiness of Severi varieties on Enriques surfaces

TL;DR

This paper proves the existence of certain nodal curves on general polarized Enriques surfaces, solving a classical problem and confirming a recent conjecture under specific conditions.

Contribution

It establishes the nonemptiness of Severi varieties of irreducible nodal curves on general Enriques surfaces with non-2-divisible polarization.

Findings

Existence of regular components of Severi varieties for all relevant δ

Solves a longstanding open problem in algebraic geometry

Confirms a recent conjecture of Pandharipande--Schmitt under non-2-divisibility condition

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 $\delta$-nodal curves in the linear system $|L|$, with $0\leq \delta\leq p_a(L)-1$. 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.