Questions on Singularities and Adjoint Linear Systems

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

Investigators

Abstract

The principal investigator plans to work on several questions, of different degrees of difficulty, concerning local and global aspects in the study of algebraic varieties. It has been understood for a long time that even if one is interested in studying smooth geometric objects, singular ones naturally appear. It is thus important to be able to measure singularities, and this is done via various invariants that can be either numerical or can have a more complicated structure. The projects in this proposal are concerned with some of these invariants and with their applications to the study of algebraic varieties. These geometric objects can be studied in two contexts: in characteristic zero, which is the more familiar context, closer to our usual Euclidean space, and in positive characteristic, a context that is closer to arithmetic. However, these two settings have close connections, and sometimes one can prove results in characteristic zero by proving the corresponding results in positive characteristic. At this point, however, most of the results concerning the global properties of algebraic varieties are only known in characteristic zero. Several of the projects in this proposal are concerned with the extension of such results to positive characteristic and with the study of the more subtle phenomena that appear in this setting. The common feature of the questions addressed in this proposal is the study of the connections between different invariants of singularities and the use of such invariants in the study of the global properties of algebraic varieties. The PI plans to explore relations between different points of view on singularities, such as the connection between spaces of arcs and the Hilbert-Samuel multiplicity and the connection between spaces of arcs and the Decomposition Theorem. He also plans to work on questions regarding the positivity properties of adjoint linear systems in positive characteristic. Such results are already known in characteristic zero, the fundamental tool in the proofs being Kodaira's vanishing theorem and its generalizations. The main difficulty in extending these results to positive characteristic is the failure of vanishing theorems in the new setting. However, new tools based on the Frobenius homomorphism have been developed in the past few years. The PI plans to use some of these tools (for example, his work with Schwede on Seshadri constants in positive characteristic) to attack several problems in this setting.