Unirationality of varieties described by families of projective hypersurfaces

TL;DR

This paper proves that under certain conditions on the dimension and degree, a family of hypersurfaces in projective space is unirational, extending previous results in algebraic geometry.

Contribution

It establishes new unirationality criteria for families of hypersurfaces with singularities, generalizing earlier findings.

Findings

Unirationality holds for large enough ambient dimension n.

Results extend previous work to broader classes of hypersurfaces.

Provides conditions relating dimension, degree, and singular locus for unirationality.

Abstract

Let $\mathscr{X}\to W$ be a flat family of generically irreducible hypersurfaces of degree $d\geq 2$ in $\PP^n$ 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 $\mathscr{X}$ is unirational. This extends results in \cite {Pre, HMP}.