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This is a Quantitative Study study.

September 28, 2025Open Access

Geometry of Hypersurfaces with Isolated Singularities

Key Points

The study shows that the variety of lines through a singular point is irreducible, providing key insights into singular hypersurfaces.
Key evidence revealed includes the examination of cubic hypersurfaces with multiple singularities, leading to significant findings on their Fano varieties.
Using geometric analysis on Fano varieties, the implications of the singularity structure are explored in depth across various constructions.
Overall, the findings may enable further research into Fano varieties and their properties, especially concerning singular hypersurfaces.

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 of lines passing through a singular point y on a hypersurface Y Pⁿ. 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.

Geometry of Hypersurfaces with Isolated Singularities

Key Points

Abstract

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