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\leq i\leq 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$(P_i)\leq d$ and rank (strength) $r_{nc}(\bar P)\geq \gamma (d,c) (1+m)^{\gamma (d,c)}$.