On Punctual Quot Schemes for Algebraic Surfaces

Let S be a smooth projective surface over the field of complex numbers C. Fix a closed point s ∈ S and a pair of positive integers r, d. By results of Grothendieck (cf. [6], [11]) there exists a projective scheme Quot [s] (r, d) parametrizing all quotient sheaves O ⊕r S → A of length d supported at s. We consider this scheme with its reduced scheme structure and call it the punctual Quot scheme. Note that Quot [s] (1, d) is nothing but the punctual Hilbert scheme Hilb d [s] studied by Briançon and Iarrobino in [1], [9]. The main result of this paper is the following extension of their results to the case r > 1: Main Theorem. Quot [s] (r, d) is an irreducible scheme of dimension (rd − 1). We prove this by exibiting a dense open subset in Quot [s] (r, d) isomorphic to a rank (r − 1)d vector bundle over Quot [s] (1, d) = Hilb d [s]. One can show that, for a quotient O ⊕r → A as above, the d-th power of the maximal ideal m S,s acts trivially on A. Hence the punctual Quot scheme Quot [s] (r, d) does not depend on S and in our proof we can assume that S = C 2. In this case a straghtforward generalization of Nakajima's construction for Hilbert schemes allows to prove the result. Remark. The original results of Briançon and Iarrobino were used by Göttsche and Soergel in [8] to show that the natural map π : Hilb d (S) → Sym d (S) is strictly semismall with respect to the natural stratifications. This leads to a simple proof of Göttsche's formula for the Poincaré polynomials of Hilb d (S). Similarly, the Main Theorem above can be used to show that the natural map π : M G (r, d) → M U (r, d) from the Gieseker moduli space of stable rank r sheaves to the Uhlenbeck compactification of the instanton moduli space, is also strictly semismall (at least in the coprime and unobstructed case). This allows one to find a connection between some homological invariants of these spaces. A systematic treatment of this questions will appear in the author's forthcoming paper. Acknowledgments. This work was originally motivated by a conjecture due to V. Ginzburg on semismallness of the map π : M G (r, n) → …