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Finding interesting curves on three-dimensional algebraic manifolds: Some special properties and interesting examples
We begin with a discussion of a very special example, a one-complex-parameter family X t of quintic hypersurfaces in complex projective 4-space, called the "Dwork pencil." These hypersurfaces each admit a one-parameter family of lines, studied by Bert van Geemen, Sheldon Katz and Alberto Albano, and Anca Mustata. There are two remarkable facts about this example: 1) If a point in projective space lies on only one hypersurface X t of the Dwork pencil, and if there is a line with contact of order 5 with the hypersurface at that point, then that line lies entirely in the hypersurface. 2) The 'relative Hilbert scheme,' that is, the space over the t-line whose fiber at t is the lines in X t, is set of relative critical points of a very explicit (multi-valued) function. We will discuss the interplay between 1) and finding rational curves on hypersurfaces of higher degree in projective 4-space. Following Donagi-Markman we will discuss the generalization of 2) to arbitrary Calabi-Yau threefolds, thereby reducing the Hodge conjecture for these threefolds to a simply-stated problem in symplectic geometry.
Thursday, October 20, 2005, 245 Altgeld Hall, 4:00 p.m.
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