Good points and constructive resolution of singluarities

Abstract

In this paper we present a new algorithm of resolution of singularities over fields of characteristic zero, making use of invariants that come from Abhyankar's good point theory [Abl]. We also prove new properties on constructive (or algorithmic) desingularization. Let us explain what we mean by an algorithm of resolution. Consider a pair (X, W) where W is a regular variety over a base field k (of characteristic zero) not necessarily irreducible (i.e. W smooth over k), and XC_W is a closed non-empty subscheme. Call C the class of all such pairs (over different base fields k). Natural maps (~ I (X) ,W1) ~' ) (X,W) arise within this class, for instance if ~: W1--,W is a smooth map over a fixed field k, or if ~: W1--*W arises from an arbitrary change of base field. Fix now a totally ordered set (I, ~<) and suppose assigned, for each pair 7~= (X, W) of U, a function ~p: X--+I which is upper-semi-continuous and takes only finitely many values, say {~1, ,.., C~r}C--I. Let max~bp be the biggest ai and set