| Abstract |
(This is a reading seminar for two graduate students.) This talk studies the birational geometry of fibered surfaces, which are integral, projective, flat schemes of dimension 2 over a Dedekind scheme. In contrast to smooth projective curves, birational equivalence for surfaces does not imply isomorphism, which leads to the problem of understanding and selecting canonical representatives within a birational class.
We first introduce basic tools for birational surface theory, including blowing-ups, contraction, and desingularization. We then explain how intersection theory on regular surfaces is used to analyze these operations and to identify exceptional curves. This perspective naturally leads to minimal surfaces and to applications of contraction criteria in the construction of canonical models. |
