Fri, September 25, 2020
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September 2020
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4:00pm [4:00pm] Sarjick Bakshi, CMI
Date and Time: Friday 25 September, 04.00pm - 05.00pm Google Meet link: Speaker: Sarjick Bakshi, CMI Title: GIT quotients of Grassmannian and smooth quotients of Schubert varieties Abstract: The Geometric invariant theory (GIT) quotients of the Grassmannian variety and its subvarieties lead to many interesting geometric problems. Gelfand and Macpherson showed that the GIT quotient of n-points in {\mathbb P}^{r-1} by the diagonal action of PGL(r,\mathbb{C}) is isomorphic to the GIT quotient of Gr_{r,n} with respect to the T-linearized line bundle {\cal L}(n \omega_r). Howard, Milson, Snowden and Vakil gave an explicit description of the generators of the ring of invariants for n even and r=2 using graph theoretic methods. We give an alternative approach where we study the generators using Standard monomial theory and we will establish the projective normality of the quotient variety for odd n and r=2. Let r < n be positive integers and further suppose r and n are coprime. We study the GIT quotient of Schubert varieties X(w) in the Gr_{r,n} admitting semistable points for the action of T with respect to the T-linearized line bundle {\cal L}(n \omega_r). We give necessary and sufficient combinatorial conditions for w for which the GIT quotient of the Schubert variety is smooth.

5:00pm [5:30pm] Shunsuke Takagi, University of Tokyo
Date and Time: 25 September 2020, 5:30 pm IST (joining time : 5:15 pm IST - 5:30 pm IST) Google meet link: Speaker: Shunsuke Takagi, University of Tokyo Title: F-singularities and singularities in birational geometry - Part 2 Abstract: F-singularities are singularities in positive characteristic defined using the Frobenius map and there are four basic classes of F-singularities: F-regular, F-pure, F-rational and F-injective singularities. They conjecturally correspond via reduction modulo $p$ to singularities appearing in complex birational geometry. In the first talk, I will survey basic properties of F-singularities. In the second talk, I will explain what is known and what is not known about the correspondence of F-singularities and singularities in birational geometry. If the time permits, I will also discuss its geometric applications.