Explicit deformation of the horospherical variety of type $\mathrm{G}_2$

TL;DR

This paper presents two geometric methods to construct a smooth family of projective varieties that interpolates between a horospherical variety of type G2 and an orthogonal Grassmannian, exploring their derived categories.

Contribution

It introduces two simple geometric constructions of a family of varieties connecting G2 horospherical and orthogonal Grassmannian types, with a brief discussion on their derived categories.

Findings

Constructed a smooth family interpolating between G2 horospherical and OGr(2,7) varieties.

Provided geometric descriptions of the family and its fibers.

Discussed the derived category of the family.

Abstract

We give two simple geometric constructions of a smooth family of projective varieties with central fiber isomorphic to the horospherical variety of type $\mathrm{G}_2$ and all other fibers isomorphic to the isotropic orthogonal Grassmannian $\mathrm{OGr}(2,7)$ and discuss briefly the derived category of this family.