Seminar Algebra and Geometry: Jan Draisma (Universiteit Eindhoven)

Finiteness for Plücker varieties

I will introduce the notion of Plücker variety, which is a rule that assigns to a pair of a natural number p and a finite-dimensional vector space V a closed subvariety of the p-th exterior power of V. To be a Plücker variety, the rule should satisfy two axioms, one of which is functoriality in V. The Grassmannian is the smallest nonempty Plücker variety, and the class of Plücker varieties is closed under joins, tangential varieties, unions, and intersections.

I will sketch a proof that any "bounded"  Plücker variety is defined set-theoretically by equations of bounded degree, and in fact by finitely many equations up to symmetry. So far, this statement was unknown even for the first secant variety of the Grassmannian. The talk is based on joint work with Rob Eggermont, and inspired by Snowden's work on Delta-varieties.

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