Standard Bundles on a Hilbert Scheme of Points on a Surface

TLDR: In this paper, the problem of enumerative geometry of the vector bundle of rank d over H d has been studied in the context of the description of the smooth structure of the 4-manifold underlying a smooth algebraic surface.

Abstract: Let S be a smooth irreducible algebraic surface over ℂ, H d a Hilbert scheme of 0-dimensional subschemes of length d in S, dim H d = 2d, and Z d ⊂ S × H d a universal family with natural projections \(S\xleftarrow{{{\tau _d}}}{Z_d}\xrightarrow{{{\pi _d}}}{H_d}\). Fix an arbitrary divisor D on S and denote \(\varepsilon _D^d = {\pi _{d*}}\tau _d^*{O_S}(D)\). Since π d is a flat finite morphism of degree d, the sheaf e D d is in fact the vector bundle of rank d over H d . We call e D d the standard vector bundle over H d . The problem of computation of its Segre classes is connected with a number of questions of enumerative geometry. In recent times it has got applications to the description of the smooth structure of the 4-manifold underlying S — see [10].