Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces

Abstract: In this paper, we study the intersection cohomology of the moduli space of semistable one-dimensional sheaves with fixed Euler characteristic, supported in a divisor class $β$ on a smooth projective surface $S$. Assuming that this moduli space is irreducible, we prove that its intersection Betti numbers in a certain range are determined by a product formula derived from Göttsche's formula for Betti numbers of Hilbert schemes of points on $S$. As an application, for Enriques and bielliptic surfaces, we show the stabilization of intersection Betti numbers of this moduli space when $β$ is sufficiently positive. In the Enriques case, we also prove a refined stabilization result concerning the perverse Hodge numbers when the moduli space is smooth.