The Milnor Fibre of the Pfaffian and the Hilbert Scheme of Four Points on C

We study a natural Hodge module on the Hilbert scheme of four points on affine three-space, which categorifies the Donaldson–Thomas invariant of the Hilbert scheme. We determine the weight filtration on the Hodge module explicitly in terms of intersection cohomology complexes, and compute the E-polynomial of its cohomology. The computations make essential use of a description of the singularity of the Hilbert scheme as the degeneracy locus of the Pfaffian function. Introduction Let (C) be the Hilbert scheme of m points on affine three-space. By general results, (C) is known to be a separated quasi-projective scheme, equipped with a proper morphism, the Hilbert–Chow morphism, to the m-th symmetric product πm : (C ) → SC. As it is well known [13], the Hilbert scheme of points of a smooth surface is itself smooth and irreducible, and πm is a resolution of singularities of the symmetric product. These statements fail for threefolds; (C) is already singular for m = 4, and for large m, it is a highly reducible and non-reduced scheme [5]. On the other hand, recent work in supersymmetric gauge theory led to a description of the Hilbert scheme as a degeneracy locus. A choice of affine Calabi–Yau structure on C induces an embedding of (C) into a smooth quasi-projective variety Mm of dimension 2m +m, which in turn is equipped with a regular function fm : Mm → C, such that (C) = {dfm = 0} ⊂ Mm is the scheme-theoretic degeneracy locus of the function fm on the smooth variety Mm; see Proposition 3.1.1. The reduced space (C ) [m] red therefore acquires a mixed 2000 Mathematics Subject Classification. Primary 14C30, 14F25, 32S40; Secondary 32S55, 32S60.