Irreducibility of moduli of vector bundles over a very general sextic Surface

TL;DR

The paper proves that the closure of the moduli space of μ-stable rank-2 locally free sheaves with determinant $H$ on a very general sextic surface $S\subset\mathbb{P}^3$ is irreducible for all $c_2\ge27$, extending the understanding of moduli geometry on general surfaces. The authors adapt O'Grady's boundary deformation method to the sextic setting, establish connectedness for $c_2\ge27$, and perform a careful boundary stratification analysis to rule out multiple irreducible components. Key technical steps include generic smoothness via obstruction theory, non-emptiness of relevant HN-strata, and precise dimension bounds for boundary strata, ensuring that potential components meet in a controlled way and that general boundary points are smooth. The results provide an effective bound on $c_2$ guaranteeing irreducibility and contribute to the broader program of describing moduli spaces of vector bundles on high-degree hypersurfaces.

Abstract

Let $S$ be a very general smooth hypersurface of degree $6$ in $\mathbb{P}^3$. In this paper we will prove that the moduli space of $μ$-stable rank $2$ torsion free sheaves with respect to hyperplane section having $c_1 = \mathcal{O}_S(1)$, with fixed $c_2 \ge 27$ is irreducible.

Theorems & Definitions (46)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• Proposition 2.5
• Definition 2.6