Alterating singularities

Abstract

Resolution of a singularity by blowing up -a nodeRecently the problern of resolution of singularities, solved by Hironaka in 1964, was considered from a completely new, fresh point of view. We report on these exciting methods of A. J. de Jong and various results. Singularities of algebraic varietiesConsider an algebraic variety V of dimension d over some field k. For example this could be an algebraic curve given by an equation like Y 2 = X 3 . We say that a point P E V is non-singular on V if "V locally around P looks like affine space Ad ." This notion can be made more precise, for example, by the J acobian criterion on the partial derivatives of equations defining the variety, or with the help of the structure of the local ring Ov,P ( the ring of germs of functions on V regular at P). Algebraic varieties show up in many disguises (solutions of equations in number theory, spaces where you like to compute integrals, solutions of differential equations, and so on). In practice we see that abstract theory works well on a non-singular variety (e.g., think of Hodge theory), and computations are usually easy to perform when working on regular models. But singularities often cause difficulties. Hence the natural question arises: given a variety V, can we find a variety V', with a map cp : V' --t V such that V' is non-singular, and cp is an isomorphism almost everywhere? This is the famous problern of resolution of singularities. Algebraic curvesHere is a case where the solution of the problern is not difficult (and classically well known). Let V = C be an algebraic curve. The construction of a non-singular model of the function field of C can be produced in several ways.(a) Here is an algebraic method. If R is is the ring of functions on an algebraic curve, and this ring is integrally closed in its field of fractions Q(R) = L, then the curve is nonsingular. Integrally closed ("normal") means that elements in L which are a zero of a monic DMV-Mitteilungen 4/ 99 equation over R are already in R. Taking the integral closure (an algebraic operation) of rings of functions on an algebraic curve produces a new curve which is non-singular. Here is an example: the affine curve, and the parametrization A 1 = C' --t C given by t r+ (x = t 2 , y = t 3 ) E C is the desingularization of this curve. The general case of an arbitrary curve is not very much more complicated. (b) Here is a geometric method. Let C C An be an affine curve, singular at P. Perform a "blowingup" of An at a center containing P. One might hope that taking something like the inverse image of C under a blowing-up (algebraic geometers say: "taking the strict transform") reduces t he singular behavior of the curve. One can see that this is indeed the case by a proper choice of the center of blowing up, and we can arrive at a desigularization after a finite number of steps. See [7, page 37, Exercise 5.6] for an explanation in easy cases.Resolution of a singularity by blowing up -a cusp 13