Deforming nonnormal isolated surface singularities and constructing 3-folds with $\mathbb{P}^1$ as exceptional set

TL;DR

This paper explores the deformation of nonnormal isolated surface singularities to construct interesting normal threefold singularities, revealing new insights into their structure and the role of hyperplane sections.

Contribution

It demonstrates how deforming nonnormal singularities can produce normal threefold singularities with specific properties, expanding the understanding of surface and threefold singularities.

Findings

Construction of non Cohen-Macaulay threefold singularities

Identification of conditions for small contractions of rational curves

Analysis of hyperplane sections of nonnormal singularities

Abstract

Normally one assumes isolated surface singularities to be normal. The purpose of this paper is to show that it can be useful to look at nonnormal singularities. By deforming them interesting normal singularities can be constructed, such as isolated, non Cohen-Macaulay threefold singularities. They arise by a small contraction of a smooth rational curve, whose normal bundle has a sufficiently positive subbundle. We study such singularities from their nonnormal general hyperplane section.