Parallel Kustin--Miller unprojection with an application to Calabi--Yau geometry

TL;DR

This paper develops a parallel unprojection theory for Gorenstein rings, enabling multiple unprojections simultaneously, and applies it to construct new families of Calabi--Yau threefolds in algebraic geometry.

Contribution

It introduces a novel parallel unprojection framework applicable to series of unprojections where ideals are pre-existing, facilitating complex geometric constructions.

Findings

Developed a new parallel Kustin--Miller unprojection theory.

Constructed 7 families of Calabi--Yau 3-folds with high codimensions.

Enhanced methods for explicit birational geometry constructions.

Abstract

Kustin--Miller unprojection constructs more complicated Gorenstein rings from simpler ones. Geometrically, it inverts certain projections, and appears in the constructions of explicit birational geometry. However, it is often desirable to perform not only one but a series of unprojections. The main aim of the present paper is to develop a theory, which we call parallel Kustin--Miller unprojection, that applies when all the unprojection ideals of a series of unprojections correspond to ideals already present in the initial ring. As an application of the theory, we explicitly construct 7 families of Calabi--Yau 3-folds of high codimensions.