Tom & Jerry triples with an application to Fano 3-folds

TL;DR

This paper introduces a novel unprojection method to construct Gorenstein rings of codimension 6, leading to new families of Fano 3-folds in weighted projective space, expanding algebraic geometry tools.

Contribution

It develops a new unprojection technique based on parallel unprojection of Kustin-Miller type, applied to construct codimension 6 Gorenstein rings and Fano 3-folds.

Findings

Constructed new Gorenstein rings of codimension 6

Produced two families of Fano 3-folds in weighted projective space

Extended unprojection theory with a novel parallel approach

Abstract

Unprojection is a theory due to Reid which constructs more complicated rings starting from simpler data. The idea of unprojection is intended for serial use. Papadakis and Neves developed a theory of parallel unprojection. In the present work we develop a new method of unprojection. Starting from a codimension 3 ideal defined by the pfaffians of a 5x5 skewsymmetric matrix, we use parallel unprojection of Kustin-Miller type in order to construct Gorenstein rings of codimension 6. We give two applications. These are two families of codimension 6 Fano 3-folds, in weighted projective space.