Log singularities, discrepancies, and thresholds with applications

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DMS-0400832ShokurovThis award supports a project of Professor Shokurov. The proposed research deals with certain singularities of algebraic varieties and log pairs, including ones that appear in the Log Minimal Model Program (LMMP). Their most fundamental characteristics, such as discrepancies, thresholds, and complements, are investigated with an aim to apply them to the LMMP in dimensions 4 and higher. This process is set forth by some new and old conjectures. PI Shokurov intends to finish the log termination for 4-folds, which completes the LMMP in dimension 4, and to obtain new results in dimensions 5 and higher. It is expected that some fundamental results on singularities must precede theLMMP, but other results, more advanced, could be interwoven. It is proposed to clarify this situation and the relations between known and new concepts, methods, and conjectures in the field of algebraic geometry toward better understanding of the LMMP within its environment. The focus will be on discrepancies, thresholds, lengths of extremal rays, the Alexeev-Borisovs and acc type conjectures, and on confinement of saturated linear systems.This is research in the field of algebra with methods and applications in algebraic geometry. A termination for flips or flops, an important class of standard transformation of geometrical objects, means finiteness of any sequence of those transformations, and effectiveness of corresponding geometrical algorithm from the computational point of view. Algebra and algebraic geometry are very old, traditional areas of modern mathematics, but which have had a revolutionary flowering in the past decades. In its origin, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials, or be given in the 3-space by the simplest geometric constructions, e.g., conic sections. Algebra is about these equations. Both fields interacts with most of branches of mathematics, e.g., analysis, topology and mathematical physics, with applications in those fields as well as in number theory, physics, discrete and computational mathematics, and robotics.