On the codimension of the singular locus

Abstract

Let $k$ be a field and $V$ an $k$-vector space. For a family $\bar{P}=\{P_i{\}}_{1\le i\le c},$ of polynomials on $V$, we denote by ${\mathbb{X}}_{\bar{P}}\subset V$ the subscheme defined by the ideal generated by $\bar{P}$. We show the existence of $\gamma (c,d)$ such that the varieties ${\mathbb{X}}_{\bar{P}}$ are smooth outside of codimension $m$, if deg$(P_i)\le d$ and rank (strength) $r_{nc}(\bar{P})\ge \gamma (d,c)(1+m{)}^{\gamma (d,c)}$.

Small corrections, added low char case