Smoothing cones over K3 surfaces

TL;DR

This paper characterizes when the affine cone over a general primitively polarized K3 surface is smoothable, revealing it occurs precisely for certain genera, and explores special singularity cases.

Contribution

It provides a complete classification of smoothability of affine cones over K3 surfaces based on genus, including examples with unique singularity behaviors.

Findings

Affine cones over K3 surfaces are smoothable if and only if genus g ≤ 10 or g=12.

Examples of singularities where affine cones are smoothable but projective cones are not.

New insights into the relationship between surface genus and cone smoothability.

Abstract

We prove that the affine cone over a general primitively polarised K3 surface of genus g is smoothable if and only if g\le 10 or g=12. We also give several examples of singularities with special behaviour, such as surfaces whose affine cone is smoothable even though the projective cone is not.