Geometry of Hypersurfaces with Isolated Singularities

TL;DR

The paper extends the study of Fano varieties of lines from smooth to mildly singular hypersurfaces. It analyzes lines through a singular point of a hypersurface, giving precise criteria for when the associated line-geometry $\Sigma$ is smooth, finite, or irreducible, depending on $d$, $m$, and $n$. It further examines cubic hypersurfaces intersected with linear spaces to produce $Y_{2r}$ with controlled isolated singularities, proving that the associated line-geometry at a singular point $y$, namely $\Sigma_y$, is irreducible of dimension $2r-2$ for $r\ge 2$. The results rely on classical tools like Bertini and Fulton–Hansen, and connect to Voisin-type maps and Calabi–Yau phenomena in the broader context of hyperplane sections and complete intersections. Overall, the work broadens the understanding of line configurations on singular hypersurfaces and their irreducibility properties, with implications for moduli and rational maps arising from such line structures.

Abstract

This paper explores the Fano variety of lines in hypersurfaces, particularly focusing on those with mild singularities. Our first result explores the irreducibility of the variety $Σ$ of lines passing through a singular point $y$ on a hypersurface $Y \subset \mathbb{P}^n$. Our second result studies the Fano variety of lines of cubic hypersurfaces with more than one singular point, motivated by Voisin's construction of a dominant rational self map.

Theorems & Definitions (33)

• Definition 2.1: Line Bundles
• Definition 2.2: Linear Equivalence
• Definition 2.3: Linear Systems and Base Loci
• Definition 2.4: Multiple Point $\mathbb{A}^n$
• Definition 2.5: Multiplicity in $\mathbb{P}^n$
• Definition 2.6: Degree of Subvariety
• Theorem 2.7: Bertini theorem
• Corollary 2.8
• Theorem 2.9: Fulton-Hansen connectedness theorem
• Remark 2.10