Resolving singularities with Cartan’s prolongation

Abstract

A standard method for resolving a plane curve singularity is the method of blow-up. We describe a less-known alternative method which we call prolongation, in honor of Cartan’s work in this direction. This method is known to algebraic geometers as Nash blow-up. With each application of prolongation the dimension of the ambient space containing the new “prolonged” singularity increases by one. The new singularity is tangent to a canonical plane field on the ambient space. Our main result asserts that the two methods, blow-up and prolongation, yield the same resolution for unibranched singularities. The primary difficulties encountered are around understanding the prolongation analogues of the exceptional divisors from blow-up. These analogues are called critical curves. Most of the critical curves are abnormal extremals in the sense of optimal control theory as it applies to rank 2 distributions (2 controls).