Singularity of cubic hypersurfaces and hyperplane sections of projectivized tangent bundle of projective space

TL;DR

This paper characterizes the singularities of cubic hypersurfaces and provides a linear algebraic description of hyperplane sections of the projectivized tangent bundle of projective space, extending previous results.

Contribution

It establishes conditions for canonical singularities of cubic hypersurfaces and describes hyperplane sections of tangent bundles in a unified framework.

Findings

Normal points of cubic hypersurfaces have canonical singularities unless they are cones over elliptic curves.

Provides a linear algebraic description of hyperplane sections of the tangent bundle of projective space.

Computes the Chow ring of these hyperplane sections.

Abstract

We show that the normal points of a cubic hypersurface in projective space have canonical singularities unless the hypersurface is an iterated cone over an elliptic curve. As an application, we give a simple linear algebraic description of all the hyperplane sections of projectivized tangent bundle of projective space, hence describing hyperplane sections of a rational homogeneous manifold of Picard rank $2$. This also simplifies and extends recent results of Mazouni-Nagaraj in higher dimensions. We also compute the Chow ring of these hyperplane sections.