Fano 4-folds in low codimension

Algebraic geometry is the study of geometric objects, called algebraic varieties, which can be described as solutions of multivariate polynomial equations. Broadly speaking, we can classify algebraic varieties in three broad classes: Fano varieties (positively curved), Calabi--Yau varieties (flat) and varieties of general type (negatively curved). It has been proved that there are only finitely many deformation families of smooth Fano varieties in any dimension. The study of mildly singular Fano varieties, called terminal Fano varieties, are of important due to their various interesting properties. In particular they form a bounded family of varieties in any dimension and they are minimal models in dimension three. In this project, we will construct families of Fano 4-folds with at worst terminal cyclic quotient singularities which can be described by the relatively low codimension embedding in some weighted projective space. In particular, we will construct smooth and terminal Fano 4-folds in codimension less than or equal to 4 as regular pullbacks from key varieties by using codimension two complete intersections, codimension three Gr(2,5) format and codimension four P2 x P2 format. Our tools will include detailed computer assisted searches, theoretical analysis of these outputs, and combination of theoretical and computational prowess to proves the existence of such geometric structures with the desired properties.