Terminal Fano four folds in low codimension

TL;DR

This paper classifies low codimension terminal Fano fourfolds with isolated orbifold points, providing a detailed description of their embeddings and a comprehensive list of families using computational methods.

Contribution

It introduces a classification of terminal Fano 4-folds in low codimension, including their embeddings and a novel algorithmic approach utilizing computer algebra.

Findings

Classified 95 families with empty or large linear systems

Classified 32 families with non-isolated Calabi-Yau sections

Developed an algorithmic method with computer algebra tools

Abstract

We construct well-formed and quasismooth terminal Fano 4-folds of index 1 in low codimension containing at worst isolated orbifold points. We provide a certain classification of these varieties where their images under the anitcanonical embedding can be described as codimension 2, 3, or 4 subvarieties of some weighted projective space. In particular, we focus on isolated terminal Fano 4-folds that either have an empty linear system or a relatively large one, but whose linear section is not an isolated canonical Calabi--Yau 3-fold. In total, we classify 95 families of terminal Fano 4-folds of the first type and 32 families of the second type. We also describe our algorithmic approach and the pivotal role of computer algebra in our results.