Unirationality of varieties described by families of projective hypersurfaces

Abstract

Let $\mathcal{X}\to W$ be a flat family of generically irreducible hypersurfaces of degree $d\ge 2$ in ${\mathbb{P}}^n$ with singular locus of dimension $t$, with $W$ unirational of dimension $r$. We prove that if $n$ is large enough with respect to $d$, $r$ and $t$, then $\mathcal{X}$ is unirational. This extends results by J. Harris, B. Mazur and R. Pandharipande in [Duke Math. J. 95 (1998), 125–160] and A. Predonzan in [Rend. Sem. Mat. Univ. Padova 31 (1961), 281–293].