Terminal Fano four folds in low codimension

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.