On the Picard numbers of moduli spaces of one-dimensional sheaves on surfaces
Fei Si, Feinuo Zhang
TL;DR
This paper investigates the Picard numbers of moduli spaces $M_{\beta,\chi}$ of one-dimensional stable sheaves supported in a large divisor class on a smooth projective surface. It establishes an asymptotic lower bound $\rho(M_{\beta,\chi}) \ge \rho(S) + 1$ under a positivity condition $(\mathrm{P})$ on $\beta$ and coprimality of $\beta\cdot H$ with $\chi$, via determinant line bundles and carefully constructed testing curves; it further relates divisor classes to relative Hilbert schemes and moduli of $\delta$-stable pairs, providing a pathway to potential upper bounds and stabilization results in several cases (K3, del Pezzo). The paper also explores wall-crossing mechanisms for stable pairs, and discusses when asymptotic irreducibility in rank-zero cases can fail, providing explicit examples on irregular surfaces. Additionally, it outlines several conjectural directions and questions, including a relative Hilbert–Chow perspective and Brill–Noether-type loci, aimed at a deeper understanding of the divisor class groups and their relation to ambient surface geometry.
Abstract
Motivated by asymptotic phenomena of moduli spaces of higher rank stable sheaves on algebraic surfaces, we study the Picard number of the moduli space of one-dimensional stable sheaves supported in a sufficiently positive divisor class on a surface. We give an asymptotic lower bound of the Picard number in general. In some special cases, we show that this lower bound is attained based on the geometry of moduli spaces of stable pairs and relative Hilbert schemes of points. Additionally, we discuss several related questions and provide examples where the asymptotic irreducibility of the moduli space fails, highlighting a notable distinction from the higher rank case.