Seminar Algebra and Geometry: Elisa Gorla (University of Neuchâtel)
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.
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