Serre’s Condition R k for Sums of Geometric Links

Abstract

We investigate what it means that the intersection of a variety with a residual intersection has a low-dimensional singular locus. For schemes having Cohen-Macaulay residual intersections, we prove, for instance, that if the intersection of the scheme with one of its geometric residual intersections has a ‘small’ singular locus, then the scheme can be defined by ‘few’ equations locally.