Seminar Algebra and Geometry - Spring 2020
|Actions of Cremona groups on CAT(0) cube complexes.||A key tool to study the plane Cremona group is its action on a hyperbolic space. Saddly, in higher rank such an action is not available. Recently in geometric group theory, actions on CAT(0) cube complexes turned out to be a powerfull tool to study a large class of groups. In this talk, based on a common work with Christian Urech, we will construct such complexes on which Cremona groups of rank n act. We will then see which kind of results on these groups we can obtain.|
no talk (Fastnacht)
|Commutative ind-groups are nested (following Cantat-Regeta-Xie)||Cantat, Regeta and Xie have recently shown that a commutative ind-group generated by an irreducible variety containing the identity is an algebraic group. We will give a short proof of this result based on the work of Furter and myself on ind-groups. This result has the immediate application that any connected affine variety whose automorphism groups is isomorphic to the automorphism group of affine $n$-space is isomorphic to affine $n$-space.|
no talk (day after Eastern)
University of the Basque Country (UPV/EHU)
|Blowup algebras of sparse determinantal varieties||Let X be a sparse generic matrix, i.e. a matrix whose entries are either zeros or distinct variables. A sparse determinantal variety is the locus where X does not have full rank. While determinantal varieties, i.e. degeneracy loci of matrices whose entries are distinct variables with no zeros, are in many respects well-understood, this is not yet the case for sparse determinantal varieties. However, sparse determinantal varieties have recently received increased attention, as new approaches for studying them have been introduced by Boocher (2011) and by Conca, De Negri and myself in a series of works (since 2015).|
Blowup algebras - such as the Rees algebra, the special fiber ring, and the associated graded ring - are an active area of study within commutative algebra. They are algebraic objects related to the concept of blowing up a variety along a subvariety. In this talk, I will present some new results on the blowup algebras of sparse determinantal varieties. Our approach makes an essential use of the theory of SAGBI bases, which I will introduce during the talk. The new results that I will present are part of an ongoing joint work with E. Celikbas, E. Dufresne, L. Fouli, K.-N. Lin, C. Polini, and I. Swanson.