A Division Theorem for Nodal Projective Hypersurfaces

Abstract

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in ${\mathbb{P}}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G^{\prime }=S{L}_{n+1}(\mathbb{C})\to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\ge 3$, and $(n,d)\ne (2,3)$. As a result, the Leray-Serre spectral sequence of the map from $V_{n,d}$ to the homotopy quotient $(V_{n,d}{)}_{h{G}^{\prime }}$ degenerates at $E_2$, and so does the Leray spectral sequence of the quotient map $V_{n,d}\to V_{n,d}/G^{\prime }$ provided the geometric quotient $V_{n,d}/G^{\prime }$ exists. We show that the latter is the case when $d>n+1$.

7 pages, more minor changes