Singularities in Algebraic Geometry

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

Investigators

Abstract

The proposal deals with invariants of singularities with the origin in birational geometry, D-module theory and tight closure theory. Invariants like the log canonical threshold or minimal log discrepancies play an important role in Mori Theory: as shown by Shokurov, their conjectural properties would have strong implications to one of the main open problems (the so-called Termination of Flips). On the other hand, it has been realized that these invariants are closely related with invariants from other fields, for example with the Bernstein polynomial (from D-module theory), with invariants defined in positive characteristic, such as the F-pure threshold, or with spaces of arcs and jets. The main goal of this proposal is to further study the connections between these different approaches, and to apply this towards a better understanding of these invariants. One of the long-term goals in algebraic geometry is to give some sort of classification of nonsingular varieties. A basic insight in the last twenty-five years is that singular spaces appear naturally into the picture, and that a good understanding of their singularities is crucial for the classification process. General properties of the invariants that measure how bad the singularities are play an important role in this study. The plan is to use approaches from various perspectives to shed some light on these general properties.