The Western Algebraic Geometry Seminar
Speakers: Sándor Kovács and James McKernan
Title: Recent advances in the Minimal Model Program, after Shokurov, I-II
One of the major discoveries of the last two decades in algebraic
geometry is the realization that the theory of minimal models of
surfaces can be generalized to higher dimensional varieties. The major
initial architects of the resulting theory in the 1980s were
Y. Kawamata, J. Kollár, S. Mori, M. Reid, and V. V. Shokurov. They
have built a theory of minimal models that works in all dimensions
except for one crucial step: the existence and termination of flips.
Flips are birational operations that only appear in higher dimensions
and their definition does not assure their existence. Nevertheless
they are essential to obtaining mimimal models. It has proved
extremely difficult to show the existence of flips. Mori proved their
existence in dimension three, which earned him the Fields Medal in
1990, but there has been very little advance in dimensions four and
higher for a long time.
Recently Shokurov introduced revolutionary new ideas that immediately
gave a more theoretical proof of the three-dimensional case and may
lead to a complete solution to the problem.
In the first talk, the Minimal Model Program will be introduced,
including key definitions, theorems, and procedures. Flips will be
defined and their importance discussed.
In the second talk, Shokurov's new ideas will be discussed and put
into perspective with regard to the previous ideas of the theory.