26 sep 2017 -- 15:00
Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
As toric varieties are well understood due to their rich combinatorial structure, a toric degeneration allows to deduce properties of the original variety. For Grassmannians, such degenerations can be obtained from birational sequences and the tropical Grassmannian. The first were recently introduced by Fang, Fourier, and Littelmann. They originate from the representation theory of Lie algebras and algebraic groups. In our case, we use a sequence of positive roots for the Lie algebra sln to define a valuation on the homogeneous coordinate ring of the Grassmannian. Nice properties of this valuation allow us to define a filtration whose associated graded algebra (if finitely generated) is the homogeneous coordinate ring of the toric variety. The second was defined by Speyer and Sturmfels and is an example of a tropical variety: a discrete object (a fan) associated to the original variety that shares some of its properties and in nice cases, as the one of Grassmannians, provides toric degenerations. In this talk, I will briefly explain the two approaches and establish a connection between them.