On the codimension of the singular locus

David Kazhdan, Tamar Ziegler

Published 2019-07-26Version 1

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 $(\{ P_i\}_{1\leq i\leq c})$. We show the existence of $\gamma (c,d)$ such that varieties $\mathbb X _{\bar P}$ are smooth outside of codimension $m$, if $deg(P_i)\leq d$ and $r(\bar P)\geq \gamma (d,c) (1+m)^{\gamma (d,c)}$, under the condition that either $char (k)=0$ or $char (k)>\max (d, |\bar d|)$, where $|\bar d| = \sum_{i=1}^c (d_i-1)$.