A remark on deformations of $1$-convex manifolds with exceptional curves

TL;DR

This paper derives a formula for the dimension of the smoothing component of certain three-dimensional singularities and applies it to specific 1-convex threefolds with exceptional curves.

Contribution

It introduces a new formula for the smoothing component dimension of 3D Cohen--Macaulay singularities and applies it to 1-convex threefolds with exceptional curves.

Findings

Derived a formula for the smoothing component dimension of Cohen--Macaulay singularities.

Applied the formula to 1-convex threefolds with exceptional curves.

Connected the geometry of exceptional curves to singularity smoothing.

Abstract

A formula for the dimension of the smoothing component of a $3$-dimensional isolated Cohen--Macaulay singularity is shown. We apply this formula for a $1$-convex threefold with a connected exceptional curve which is blown down to a terminal Gorenstein singularity.