Fitting ideals and multiple-points of surface parameterizations

TL;DR

This paper explores how to explicitly describe multiple-point sets on algebraic surfaces parameterized birationally, using Fitting ideals of graded parts of the symmetric algebra.

Contribution

It introduces a novel method to characterize multiple points on surfaces via Fitting ideals derived from the symmetric algebra of the parameterization.

Findings

Explicit descriptions of multiple-point sets in terms of Fitting ideals

Connection between algebraic properties and geometric features of surfaces

Method applicable to various classes of algebraic surface parameterizations

Abstract

Given a birational parameterization of an algebraic surface S in the projective space, the purpose of this paper is to investigate the sets of points on S whose preimage consists in k or more points, counting multiplicities. They are described explicitly in terms of Fitting ideals of some graded parts of the symmetric algebra associated to this parameterization.