Unirationality of varieties described by families of projective hypersurfaces

Let \mathscr{X}\to W be a flat family of generically irreducible hypersurfaces of degree d\geq 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 \mathscr{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].