A Division Theorem for Nodal Projective Hypersurfaces

TL;DR

This paper proves a division theorem for the cohomology of the space of hypersurfaces with simple nodal singularities, showing the surjectivity of the orbit map and spectral sequence degeneration under certain conditions.

Contribution

It establishes a surjectivity result for the orbit map on rational cohomology and demonstrates spectral sequence degeneration for hypersurfaces with simple nodes, extending understanding of their geometric and topological structure.

Findings

Orbit map is surjective on rational cohomology for specified hypersurfaces.

Spectral sequences degenerate at E2 under certain conditions.

Quotient space exists when degree exceeds dimension plus one.

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 $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence of the map from $V_{n,d}$ to the homotopy quotient $(V_{n,d})_{hG'}$ degenerates at $E_2$, and so does the Leray spectral sequence of the quotient map $V_{n,d}\to V_{n,d}/G'$ provided the geometric quotient $V_{n,d}/G'$ exists. We show that the latter is the case when $d>n+1$.