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 \(G'=SL_{n+1}({\mathbb {C}}) \rightarrow V_{n,d}\), \(g\mapsto g\cdot s_0\), \(s_0\in V_{n,d}\) 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 \((V_{n,d})_{hG'}\) degenerates at \(E_2\), and so does the Leray spectral sequence of the quotient map \(V_{n,d}\rightarrow V_{n,d}/G'\) provided the geometric quotient \(V_{n,d}/G'\) exists. We show that the latter is the case when \(d>n+1\).