The Cramer Varieties Cr(r,r+s,s)

TL;DR

This paper investigates quasi-homogeneous affine algebraic varieties, focusing on their tangent bundles and canonical classes through group representations, with a special case linked to the Orthogonal Grassmannian OGr(5,10).

Contribution

It provides a method to describe tangent bundles and canonical classes of these varieties using group representations, including a notable example connected to OGr(5,10).

Findings

Describes tangent bundle structure in terms of group representations.

Expresses canonical class of quasi-homogeneous varieties.

Links specific case to Orthogonal Grassmannian OGr(5,10).

Abstract

In this paper we study quasi-homogeneous affine algebraic varieties, that is, varieties obtained as closures of orbits of suitable group representations. We also discuss one interesting case that has links with the Orthogonal Grassmannian OGr(5,10). The main aim is to write the tangent bundle and the canonical class of quasi-homogeneous affine algebraic varieties in terms of group representations.