UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

October 14, 2014

939 Evans Hall

3:45PM: Families of fibrations of del Pezzo surfaces

Brendan Hassett

Our goal is to understand rational points of del Pezzo surfaces over function fields of curves. In geometric terms, consider fibrations in quartic del Pezzo surfaces over P^1 with mild singular fibers. There is a fundamental numerical invariant of such a family, known as the height. We show that the family of all del Pezzo surfaces of fixed height is non-empty and irreducible, with the exception of a few small heights. We describe how the spaces of sections of these fibrations map to their intermediate Jacobians. (joint with Kresch and Tschinkel)

5:00PM: Normality of Secant Varieties

Brooke Ullery

If X is a smooth variety embedded in projective space, we can form a new variety by looking at the closure of the union of all the lines through 2 points on X. This is called the secant variety to X. Similarly, the Hilbert scheme of 2 points on X parametrizes all length 2 zero-dimensional subschemes. I will talk about how these two constructions are related. More specifically, I will show how we can use certain tautological vector bundles on the Hilbert scheme to help us understand the geometry of the secant variety, leading to a proof that for sufficiently positive embeddings of X, the secant variety is a normal variety.

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