Asymptotic Regularity for Symbolic Powers of Determinantal Ideals

This chapter applies the methods of Chap. 9 to the study the ideals “defined by shape”. They allow natural filtrations that lead to cohomology computations in a characteristic-free fashion. The filtrations take a particularly nice form for symbolic powers of determinantal ideals, where the vanishing theorems from Chap. 9, combined with the characterization of Castelnuovo-Mumford regularity in Chap. 8, allow us to determine an explicit formula for the asymptotic regularity. We end Chapter 10 with a brief survey of several other homological and arithmetic properties of determinantal ideals that can be derived in a compact way via geometric arguments.