Free resolutions for multiple point spaces

TL;DR

This paper investigates the algebraic structure of multiple-point schemes for corank 1 map-germs, revealing how their resolutions relate to those of the ambient space, with implications for understanding singularities.

Contribution

It establishes a specific relationship between the presentations of multiple-point schemes and the ambient space for corank 1 germs, a phenomenon not observed in higher corank cases.

Findings

The matrix of a presentation of $ ext{O}_{D^{k+1}(f)}$ over $ ext{O}_{D^k(f)}$ is a submatrix of the ambient space's presentation matrix.

This relationship holds specifically for corank 1 germs and not for higher corank.

Provides a new perspective on the algebraic structure of multiple-point spaces in singularity theory.

Abstract

Let f be a map-germ of corank 1 from complex n-space to complex (n+1)-space, and, for k less than or equal to the multiplicity of f, let $D^k(f)$ be its k'th multiple-point scheme -- the closure of the set of ordered k-tuples of pairwise distinct points sharing the same image. There are natural projections from $D^{k+1}(f)$ to $D^k(f)$, determined by forgetting one member of the (k+1)-tuple. We prove that the matrix of a presentation of $\OO_{D^{k+1}(f)}$ over $\OO_{D^k(f)}$ appears as a certain submatrix of the matrix of a suitable presentation of $\OO_{\CC^n}$ over $\OO_{\CC^{n+1}}$. This does not happen for germs of corank greater than 1.