Punctual Hilbert schemes and resolutions of multiple point singularities

Abstract

If f : X" ~ YP is a morphism of smooth complex analytic varieties with n < p, then the multiple points of order k of f i n the target are those y E YP with k preimages, each preimage counted with proper multiplicity. It is clear that to understand the geometry o f f it is necessary to understand these multiple point sets and their closures. Much work has also been done recently to calculate characteristic class formulas for these sets and their relatives in the source off(cf . [1, 10, 17]). One approach to the problem of calculating the characteristic class formula of a singularity involves finding resolutions of the singularities [4, 15, 18]. Traditionally in singularities of maps, one uses multi-jet bundles to study and control multiple point singularities. Jet bundles often enter also in the constructions made to resolve singularities. Unfortunately, as will be discussed in detail later, multi-jet bundles are difficult to work with if not useless when the problem involves the closure of the set of multiple points. In this paper we give a new construction, using punctual Hilbert schemes, which we offer as an alternative to multi-jets in the study of multiple point singularities. As an illustration of its usefulness, we use it to find a resolution of the closure of the triple point set of any "good" map f (Theorem 2.3), and of the multiple point set of a " g o o d " f o f any order, provided the m a p f has kernel rank at most 2 (Theorem 1.6). (Recall that the kernel rank of f a t x is the dimension of the kernel of Df(x).) This construction also provides a useful starting point for finding resolutions of multiple point sets for general f, by reducing the problem of resolving the singularity of f t o the problem of resolving the singularities of the corresponding Hilbert schemes. In a later paper we will show how a modification of our construction allows one to resolve triple points in the source as well.