The 4-Intersection Unprojection Format

TL;DR

This paper introduces the 4-intersection unprojection format, a new algebraic construction in geometry that enables the creation of complex Gorenstein rings and Fano 3-folds, expanding tools for algebraic geometry research.

Contribution

It presents a novel unprojection format based on four intersections, leading to new methods for constructing codimension 6 Gorenstein rings and Fano 3-folds.

Findings

Constructed three families of codimension 6 Fano 3-folds.

Developed a new unprojection format called 4-intersection.

Enabled new algebraic geometric constructions.

Abstract

Unprojection theory is a philosophy due to Miles Reid, which becomes a useful tool in algebraic geometry for the construction and the study of new interesting geometric objects such as algebraic surfaces and 3-folds. In the present work we introduce a new format of unprojection, which we call the 4-intersection format. It is specified by a codimension 2 complete intersection ideal which is contained in four codimension 3 complete intersection ideals and leads to the construction of codimension 6 Gorenstein rings. As an application, we construct three families of codimension 6 Fano 3-folds embedded in weighted projective space.