Diptych varieties. I

TL;DR

This paper introduces a new class of affine Gorenstein 6-folds created by smoothing singularities in reducible affine toric surfaces, with applications in algebraic geometry and Mori flip models.

Contribution

It constructs a novel class of affine Gorenstein 6-folds using explicit toric geometry methods and Gorenstein unprojection techniques, expanding the toolkit for algebraic geometers.

Findings

Established existence of these varieties using explicit toric methods.

Demonstrated applications in constructing models of Mori flips.

Provided explicit examples of the new varieties.

Abstract

We present a new class of affine Gorenstein 6-folds obtained by smoothing the 1-dimensional singular locus of a reducible affine toric surface; their existence is established using explicit methods in toric geometry and serial use of Kustin-Miller Gorenstein unprojection. These varieties have applications as key varieties in constructing other varieties, including local models of Mori flips of Type A.