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

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.

💡 Research Summary

The paper by Bansal, Sarkar, and Vats studies two closely related problems in complex projective geometry: the singularities of normal points on cubic hypersurfaces and the geometry of hyperplane sections of the projectivized tangent bundle of projective space, ( \mathbb{P}(T\mathbb{P}^n) ).

Theorem A (Canonical singularities of cubic hypersurfaces).
Let (X\subset\mathbb{P}^n) be a cubic hypersurface and (U\subset X) a non‑empty open normal subset. The authors prove that, unless (X) is an iterated cone over an elliptic curve, (U) has canonical singularities. The proof proceeds by induction on the ambient dimension. For (n\le3) the statement follows from classical results. Assuming a non‑canonical point (x_0\in U) for (n\ge4), they show that every line through (x_0) and any other point of (X) must lie in (X). This forces (X) to be a cone over a cubic hypersurface (Y\subset\mathbb{P}^{n-1}) with vertex (x_0). By adjunction, (Y) inherits a non‑canonical singularity, and the induction hypothesis forces (Y) to be an iterated cone over an elliptic curve, whence (X) itself is such a cone. Consequently, any normal locus of a cubic hypersurface that is not an iterated cone has only canonical (hence rational) singularities. This result extends earlier work on cubic surfaces to arbitrary dimension and provides a clean criterion for canonical singularities in this setting.

Theorem B (Hyperplane sections of ( \mathbb{P}(T\mathbb{P}^n) )).
The authors identify the complete linear system (|\mathcal{O}{\mathbb{P}(T\mathbb{P}^n)}(1)|) with the projective space ( \mathbb{P}(W^\vee) ), where (W = M{n+1}(\mathbb{C})/) scalar matrices. For a matrix class (