K3 transitions and canonical 3-folds

TL;DR

This paper introduces K3 transitions as a geometric method to connect different classes of canonical 3-folds, providing insights into their deformation properties and singularities.

Contribution

It presents a novel geometric framework for studying canonical 3-folds through K3 transitions, linking deformation classes via birational contractions and smoothings.

Findings

Analysis of the web of canonical 3-folds in small codimension

Identification of a singularity with obstructed smoothings in codimension 4

Example of a singularity similar to the affine cone over a degree 6 del Pezzo surface

Abstract

We introduce K3 transitions as a geometric approach to studying canonical 3-folds. These transitions link different deformation classes of canonical 3-folds via a combination of birational contractions and smoothings. As applications, we investigate some basic properties of the web of canonical 3-folds in small codimension and give an interesting example of a singularity in codimension 4 with obstructed smoothings, similar to the famous example of the affine cone over a degree 6 del Pezzo surface.