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SUMMARY:Lara Bossinger (Oaxaca)
DTSTART:20200910T150000Z
DTEND:20200910T160000Z
DTSTAMP:20260423T005759Z
UID:notts_ag/24
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/notts_ag/24/
 ">Families of Gröbner degenerations\, Grassmannians\, and universal clust
 er algebras</a>\nby Lara Bossinger (Oaxaca) as part of Online Nottingham a
 lgebraic geometry seminar\n\n\nAbstract\nLet $V$ be the weighted projectiv
 e variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal co
 ne in the Gröbner fan of $J$ with m rays. We construct a flat family over
  affine $m$-space that assembles the Gröbner degenerations of $V$ associa
 ted with all faces of $C$. This is a multi-parameter generalization of the
  classical one-parameter Gröbner degeneration associated to a weight. We 
 show that our family can be constructed from Kaveh-Manon's recent work on 
 the classification of toric flat families over toric varieties: it is the 
 pullback of a toric family defined by a Rees algebra with base $X_C$ (the 
 toric variety associated to $C$) along the universal torsor $\\mathbb{A}^m
  \\to X_C$. If time permits I will explain how to apply this construction 
 to the Grassmannians $\\mathrm{Gr}(2\,n)$ (with Plücker embedding) and $\
 \mathrm{Gr}(3\,6)$ (with "cluster embedding"). In each case there exists a
  unique maximal Gröbner cone whose associated initial ideal is the Stanle
 y-Reisner ideal of the cluster complex. We show that the corresponding clu
 ster algebra with universal coefficients arises as the algebra defining th
 e flat family associated to this cone. Further\, for $\\mathrm{Gr}(2\,n)$ 
 we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be rec
 overed as tropicalized cluster mutation. This is joint work with Fatemeh M
 ohammadi and Alfredo Nájera Chávez.\n
LOCATION:https://researchseminars.org/talk/notts_ag/24/
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