K-instability of hyperplane sections of Segre Varieties

TL;DR

This paper proves that most hyperplane sections of Segre varieties are K-unstable unless they are smooth and the dimensions are equal, highlighting instability in geometric invariant theory.

Contribution

It establishes the K-instability of normal hyperplane sections of Segre varieties when they are not smooth or when dimensions differ.

Findings

Normal hyperplane sections of Segre varieties are K-unstable if not smooth.

K-unstability occurs when the dimensions m and n are not equal.

Smooth hyperplane sections with equal dimensions are not covered, implying stability in those cases.

Abstract

We prove that a normal hyperplane section of the Segre variety $\Sigma_{m, n}$ is K-unstable with respect to any polarization if $m\neq n$ or it is not smooth.