Singularities of Birational Geometry via Arcs and Differential Operators

Abstract

We study singularities of algebraic varieties, in particular those arising in birational geometry, from several points of view. The first is that of arc schemes: arc schemes parametrize “infinitesimal curves” on a variety, and their geometry reflects properties of singularities. We show that morphisms of arc schemes (more precisely, of “local” arc schemes) can detect local isomorphisms of varieties. More precisely, we use the triviality of a certain ideal-closure operation to show that if a morphism induces an isomorphism of local arc schemes then it must be an isomorphism on local rings.

We then use arc schemes, in conjunction with the theory of determinantal rings, to verify the semicontinuity conjecture for the behavior of the minimal log discrepancy (a subtle invariant of singularities) in the case of determinantal varieties. In particular, we calculate the Nash ideal of a generic square determinantal variety, which then allows us to give an explicit formula for the minimal log discrepancies of pairs of determinantal varieties and determinantal subvarieties. This allows us to verify the semicontinuity conjecture for such pairs.

We then take another point of view, via the study of differential operators on singular rings. At least since [Levasseur and Stafford 1989], the question had been asked of whether one can characterize singularities of rings via certain properties of their rings of differential operators. In particular, one question is whether a ring with mild singularities is a simple module under the action of its ring of differential operators. While an answer in characteristic p had been provided by [Smith 1995], no answer had been forthcoming in characteristic 0. We provide a counterexample showing that the expected connection does not exist, through the study of the global geometry of Fano varieties. More specifically, we show that certain del Pezzo surfaces do not have big tangent bundles, and thus their homogeneous coordinate rings are not simple under the action of their rings of differential operators, despite having “mild” singularities.