The Problem of Subtraction in Algebraic Geometry and Commutative Algebra

Abstract: Some curves in 3-space can be realized as the intersections of two surfaces; for example, the intersection of two quadric (= degree 2) hypersurfaces containing a line in common has another component, which can be thought of as the intersection minus the line. The invariants of that other component can be computed from this information: it must be a curve of degree 3 and genus 0. In commutative algebra, subtractions appear as ideal quotients, and raise other interesting questions, some very subtle. Such problems have been studied for more than 100 years. I'll discuss the origins of this theory of "residual intersections", and some of the modern developments in algebraic geometry and commutative algebra.

Mathematics

Audience: researchers in the topic

IPM-Isfahan Mathematics Colloquium