Brill-Noether loci on an Enriques surface covered by a Jacobian Kummer surface

TL;DR

The paper addresses constructing nontrivial first Brill-Noether loci for rank-2 bundles on Enriques surfaces $Y$ that are covered by Jacobian Kummer surfaces. By leveraging the double cover $\pi:X\to Y$ from a Jacobian Kummer surface $X$ and a Takemoto–Kim-type stability criterion, it shows that for suitable divisors $D$ on $X$ with $h^0(\mathcal{O}_X(D))=1$, $D^2\le -4$, and $\theta^*$-noninvariant subdivisors, the pushforward $\mathcal{V}=\pi_*\mathcal{O}_X(D)$ is a stable rank-2 bundle on $Y$ with $h^0(\mathcal{V})=1$, yielding a nonempty $W^1_H(2;c_1,c_2)$ that is strictly contained in the moduli space $M_H(2;c_1,c_2)$ for any polarization $H$. The construction provides explicit divisor families on the K3 cover (e.g., sums of nodes or node-plus-trope combinations) that satisfy the hypotheses and produce BN divisors, including a concrete example $D=E_0+E_{12}+T_{136}+T_{146}$. This work advances higher-rank Brill-Noether theory on Enriques surfaces by giving concrete, geometrically grounded criteria to realize BN divisors via line bundles on the K3 cover and their pushforwards.

Abstract

The aim of this note is to exhibit proper first Brill-Noether loci inside the moduli spaces $M_{Y,H}(2;c_1,c_2)$ of $H$-stable rank $2$ vector bundles with fixed Chern classes of a certain type on an Enriques surface $Y$ which is covered by a Jacobian Kummer surface $X$.

Theorems & Definitions (25)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• proof