Geometry of Hypersurfaces with Isolated Singularities

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 $\mathrm{\Sigma }$ 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.