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

In this paper, we develop a unified approach to study the intersection Betti numbers of moduli spaces of one-dimensional semistable sheaves on smooth projective surfaces. Assuming the irreducibility of such moduli spaces, we prove that their intersection Betti numbers in a certain range of degrees coincide with the stable Betti numbers of Hilbert schemes of points. As an application, for minimal surfaces of Kodaira dimension 0, we show that these intersection Betti numbers stabilize in each fixed degree, which fits into the broader context of stable cohomology for moduli spaces of sheaves. In the case of Enriques surfaces, we also prove a refined stabilization result related to Oberdieck’s conjecture on perverse Hodge numbers.

💡 Research Summary

The paper develops a unified framework for understanding the intersection Betti numbers of moduli spaces of one‑dimensional semistable sheaves on smooth projective surfaces. Let (S) be a smooth complex projective surface equipped with a fixed ample divisor (H). For an effective divisor (\beta) and an integer (\chi), the authors consider the coarse moduli space (M_{\beta,\chi}) parametrising S‑equivalence classes of one‑dimensional sheaves (F) with determinant (\mathcal O_S(\beta)) and Euler characteristic (\chi). The natural Hilbert–Chow morphism (h\colon M_{\beta,\chi}\to |\beta|) sends a sheaf to its Fitting support; over the open subset (|\beta|_{\mathrm{int}}) of integral curves the fibres are compactified Jacobians.

Because (M_{\beta,\chi}) is generally singular, the study is carried out in intersection cohomology (IH^k(M_{\beta,\chi})). The main theorem (Theorem 1.1) asserts that, assuming (M_{\beta,\chi}) is irreducible and that (\beta) is sufficiently positive—more precisely (\beta) is (\max{1,2k-2})-very ample—and that the degree (k) lies below both the codimension of the non‑integral locus and (\frac23\dim|\beta|), the intersection Betti numbers coincide with the stable Betti numbers (b_\infty^k) of Hilbert schemes of points on (S). The numbers (b_\infty^k) are given by the generating series \