Singularities: Geometric and Arithmetic Aspects

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The proposal concerns the study of singularities of algebraic varieties. One studies these singularities in two settings: in characteristic zero and in positive characteristic. In each setting one defines invariants that measure the singularities. Despite the different definitions, the two sets of invariants exhibit many common features and subtle connections. In characteristic zero, the invariants to be studied (more precisely, the log canonical threshold, and the minimal log discrepancies) are defined in terms of divisorial valuations. A fundamental result, the existence of resolution of singularities, gives a strong finiteness statement when computing these invariants. The motivation for the interest in such invariants comes from their applications to birational geometry: certain conjectural properties of these invariants imply the termination of flips, one of the outstanding open problems in higher dimensions. The PI plans to study questions related to the main conjecture on log canonical thresholds, that asserts that in fixed dimension, there are no increasing sequences of such invariants. Another part of the proposal concerns invariants defined in positive characteristic via the Frobenius morphism. They come out of tight closure theory, but have many subtle connections with the invariants in characteristic zero. In particular, a basic conjecture in the field connects in a precise way the invariants in characteristic zero to those in positive characteristic, via reduction mod p. The PI proposes to use D-module theoretic ideas in order to study this conjecture. Singularities are responsible for many phenomena, both in geometry and in arithmetic. For example, they govern the asymptotic of the number of solutions of congruences modulo prime powers, or the integrability of powers of polynomials. It has been realized in the past few years that invariants of singularities that show up in different contexts have deep connections amongst them. One can hope that by approaching singularities from these various points of view, one can get a richer picture, that can be used to attack problems with applications in other fields.