UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

November 04, 2014

939 Evans Hall

3:45PM: Geometry of varieties of general type

B. Purnaprajna

In this talk we will explore the geometry of varieties of general type with special emphasis on algebraic surfaces. The talk will deal with deformations of canonical maps and topology of fibered algebraic surfaces. The applications include a solution to a question of Enriques, description of components of moduli spaces, holomorphic convexity of the universal cover, and results on very ampleness and deformation of some threefolds such as Calabi Yau. The talk will be accessible to algebraists as well. If time permits, we will state some new results on deformation of varieties of general type of arbitrary dimension.

5:00PM: Hunting for secant varieties with unexpected dimensions

Hirotachi Abo

This talk is concerned with defective secant varieties (i.e., secant varieties that do not have the expected dimension). The most basic tool in the study of secant varieties is the so-called Terracini lemma. Terracini's lemma describes the tangent space to a secant variety in terms of linear algebra, and thus it makes a computer algebra system feasible to effectively compute the dimension of the secant varieties of a large variety. In particular, Terracini’s lemma can be used to verify the non-defectivity of secant varieties. It can also be used to experimentally detect potential defective secant varieties. Although the result of a computation provides strong evidence and gives a dimensional lower bound, it cannot be used as a rigorous proof of its deficiency. In conclusion, proving that computationally suggested defective secant varieties are actually defective requires more insights. In this talk, I discuss a geometric argument that proves experimentally suggested defective varieties are indeed defective. This geometric argument enabled us to find surprisingly many defective secant varieties of Segre-Veronese varieties.

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