The arithmetic rank of determinantal nullcones

Abstract

We compute the arithmetic rank as well as the local/\'etale cohomological dimension of nullcone ideals arising from the classical actions of the symplectic group, the general linear group, and the orthogonal group. We use these calculations to establish striking vanishing results for local cohomology modules supported at these nullcone ideals; this is achieved via a careful analysis of the critical local cohomology modules. The vanishing theorems that we prove are sharp in various respects.