Date and Time: Friday 25 September, 04.00pm - 05.00pm
Google Meet link:
https://meet.google.com/mvd-txng-kgf
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.