Standard Bundles on a Hilbert Scheme of Points on a 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 ε ^d_D is in fact the vector bundle of rank d over H _d . We call ε ^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].