Fano 3-folds from homogeneous vector bundles over Grassmannians

Abstract

We rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.

Topics on Fano varieties of K3 type

“…For example, approximately 80% of all Fano in dimension up to three admits a description as SR Fano. This is the point made in [DFT21], and we would expect a similar ratio in higher dimension as well.…”

“…As for the remaining 88, the original classification completed by Mori and Mukai in [MM86] relied heavily on the birational tool of Mori theory. A biregular classification can be found in [CCGK16], where every Fano 3-fold was described as either complete intersection in toric varieties or as zero loci in homogeneous varieties, and in [DFT21], where the entire classification was rewritten by considering only products of possibly weighted Grassmannians as ambient space. We refer to [Bel] for an excellent overview.…”

Polyvector fields for Fano 3-folds

“…For the Picard rank 1 case Mukai alternatively described the classification using the vector bundle method in [43], by writing Fano 3-folds of Picard rank 1 as zero loci of vector bundles on homogeneous varieties and weighted projective spaces. In higher Picard ranks this was extended in 2 different ways (which have partial overlap), by giving a description as (1) zero loci of vector bundles on GIT quotients by products of general linear groups [15]; or (2) zero loci of homogeneous vector bundles on homogeneous varieties [16].…”

“…In the second variation the key variety F is always a homogeneous variety. In particular, it is shown in [16] that every Fano 3-fold can be realised as the zero locus of a homogeneous vector bundle in a product of (possibly weighted) Grassmannians.…”

“…What is interesting to observe is that the homogeneous methods from [16] are very good at determining Hodge numbers (and in particular they are expected to help in classifying Fano 4-folds, see e.g. [8]), with only a dozen deformation families of Fano 3-folds not being fully determined.…”