Chennai Mathematical Institute


Wednesday, 21 April 2021, 3.30 pm
Torus quotients of Richardson Varieties in the Grassmannian

Sarjick Bakshi
Chennai Mathematical Institute.


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.

All are invited to attend the viva-voce examination.