Moduli spaces of one dimensional sheaves on log surfaces and Hilbert schemes

TL;DR

The paper addresses the structure of moduli spaces ${\mathcal{M}_{\beta}(X, D, {\mathfrak{d}})}$ of one-dimensional sheaves on log surfaces, showing they have symplectic singularities and admit a unique symplectic resolution when $X$ is a smooth rational surface and $D$ is anticanonical. It employs twisted Abel maps to relate local moduli data to relative Hilbert schemes ${\mathrm{Hilb}^g(\mathcal{C}/\Lambda)}$, then uses a nondegeneracy condition to produce a correspondence between relative Hilbert schemes and products of Hilbert schemes of $A_n$-type singularities. The main contribution is a precise local model: each point of ${\mathcal{M}_{\beta}(X, D, {\mathfrak{d}})}$ is formally a product of ${\mathrm{Hilb}^{d_i}(S_{w_i-1})}$ and an affine space, yielding symplectic singularities and a unique symplectic resolution. The results provide a robust framework for understanding partial compactifications of relative Jacobians in log Calabi-Yau settings and connect to Hilbert schemes of surface singularities and their symplectic geometry.

Abstract

Let $X$ be a smooth projective rational surface, $D\subset X$ an effective anticanonical curve, $β$ a curve class on $X$ and $\mathfrak{d}=\sum w_iP_i$ an effective divisor on $D_{\mathrm{sm}}$. We consider the moduli space $\mathcal{M}_β(X, D, \mathfrak{d})$ of sheaves on $X$ which are direct images of rank-$1$ torsion-free sheaves on integral curves $C$ in $β$ such that $C|_D=\mathfrak{d}$, and show that each point of $\mathcal{M}_β(X, D, \mathfrak{d})$ is smooth over a point in the product of the Hilbert schemes of surface singularities of types $A_{w_i-1}$. Hence, $\mathcal{M}_β(X, D, \mathfrak{d})$ has symplectic singularities and admits a unique symplectic resolution.

Theorems & Definitions (40)

• Theorem 1.1: =Theorem \ref{['thm_main']}
• Definition 2.1
• Theorem 2.2: Tyurin1988, Bottacin1995b
• Theorem 2.3: BiswasGomez2020, cf. Beauville1990, Bottacin1995a
• Definition 2.5
• Remark 2.6
• Lemma 2.7
• proof
• Definition 3.1
• Definition 3.2