Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces
Fei Si, Feinuo Zhang
TL;DR
This work analyzes the intersection cohomology of moduli spaces M_{β,χ} of semistable one-dimensional sheaves on a smooth projective surface S with fixed determinant and Euler characteristic. Under irreducibility and positivity assumptions on β, the authors show that the k-th intersection Betti number matches a Göttsche-type product b_k^∞ in a range, tying the moduli space’s topology to that of Hilbert schemes of points via the Hilbert–Chow morphism and perverse filtrations. They establish a robust stabilization phenomenon for Enriques and bielliptic surfaces: as β grows, IH^k(M_{β,χ}) stabilizes to b_k^∞, with a refined stabilization for perverse Hodge numbers when the moduli space is smooth (Enriques case). The paper develops a detailed framework using relative Hilbert schemes, k-very ampleness, local versality of curve families, and perverse filtrations to connect deformation theory of planar curve singularities to global invariants, yielding explicit product-form generating series for stabilized invariants and offering new insights into the geometry of these moduli spaces.
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.