Singularity of cubic hypersurfaces and hyperplane sections of projectivized tangent bundle of projective space
Ashima Bansal, Supravat Sarkar, Shivam Vats
TL;DR
The paper investigates canonical singularities of cubic hypersurfaces and the structure of hyperplane sections of the projectivized tangent bundle $\mathbb{P}(T_{\mathbb{P}^n})$. It proves that a cubic hypersurface not arising as an iterated cone over an elliptic curve has canonical singularities on its normal locus, using an induction-on-$n$ argument and a Bertini-type analysis. It then provides a linear-algebraic description of hyperplane sections $H_{[\bar{A}]}$ of $\mathbb{P}(T_{\mathbb{P}^n})$, showing reducibility occurs exactly when $[\bar{A}]$ comes from a rank-one matrix, and, in the irreducible case, that $H_{[\bar{A}]}$ is a normal rational Fano of dimension $2n-2$ with canonical singularities and two contractions to $\mathbb{P}^n$. For general $[\bar{A}]$, the sections are smooth with Picard rank $2$, and the paper also computes the Chow ring $A^*(H)$ with an explicit presentation. These results extend Mazouni–Nagaraj’s work to higher dimensions and yield a complete linear-algebraic framework for hyperplane sections of a broad class of rational homogeneous spaces of Picard rank $2$, with consequences for dual varieties and degenerations.
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.