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Seminar of Geometry and Arithmetic, Winter 2007 Fridays at 1:30pm in South Hall, Room SH 6635
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| January 12, 2007 | Abstract: Traditionally characteristic p geometry had a reputation in algebraic geometry of being harder than characteristic zero (e.g Mumford’s papers, Pathologies I-IV). However, Mori’s solution of Hartshorne’s conjecture is an example where characteristic p geometry is considerably easier. This talk will report on a recent reconaissance mission, conducted by Cascini, Hacon and myself, into the territory of Mori theory in characteristic p, which illustrates both types of behaviour in a very surprising way. |
| February 9, 2007 | Abstract: Let X be a smooth complex projective variety and a:X --> A be the Albanese map (which is given by integrating a basis of the global holomorphic 1 forms on X). By a result of Kawamata, it is known that: (i) If the Kodaira dimension of X is zero, then a:X\to A is surjective and has connected fibers. In this talk I will explain the following result (which was conjectured by Ueno and is joint work with J A Chen): (ii) If the Kodaira dimension of X is zero and F is a general fiber of the Albanese map a:X --> A, then the Kodaira dimension of F is also zero. In the case dim X =2, this means that if X is a surface of Kodaira dimension 0 then the Albanese map is either trivial or a surjection to an elliptic curve whose general fiber is an elliptic curve or birational. |
| March 9, 2007 | Abstract: I will briefly review the construction of the Hilbert scheme in order to discuss its component structure, the role played by Borel-fixed ideals, and some local results on the structure of the Hilbert scheme at the points corresponding to these ideals. |
