Algebraic Geometry seminar
Wednesday, 15 March 2023, 11.30 am
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Venue: Ramanujan Hall
Host: Saurav Bhaumik
Speaker: Pinakinath Saha
Affilaition: IIT Bombay
Title: On (weak) Fano $G$-Bott-Samelson-Demazure-Hansen varieties
Abstract: Let $G$ be a semi-simple simply connected algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $W$ be the Weyl group of $G$ with respect to $T$. For an arbitrary sequence $w$ of simple reflections in $W,$ let $Z_{w}$ be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to $w.$ Bott-Samelson-Demazure-Hansen-varieties are an important tool in geometric representation theory. They were originally defined as desingularizations of Schubert varieties and were used to describe the geometry of Schubert varieties. There is a natural action of $B$ on $Z_{w}$ given by the left multiplication. Let $\widetilde{Z_{w}}:=G\times^{B}Z_{w}$ be the fibre bundle associated to the principal $B$-bundle $G\to G/B.$ We call it $G$-Bott-Samelson-Demazure-Hansen variety ($G$-BSDH-variety for short). In the first part of the talk, we will describe a basis of the Picard group of $G$-BSDH variety, which we will refer as the $\mathcal{O}(1)$-basis. Then we will characterize the nef, globally generated, ample and very ample line bundles on $G$-BSDH variety in terms of the $\mathcal{O}(1)$-basis. Finally, we will provide a characterization of (weak) Fano $G$-BSDH varieties. We introduce a few more notations for the second part of my talk. Let $G=SO(8n,\mathbb{C})\big/SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the end simple root of its Dynkin diagram. In the second part of the talk, we will discuss the projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle. The first part of the talk will be based on joint work with Saurav Bhaumik. Here is the link of the preprint: \url{https://arxiv.org/abs/2212.10366}. The second part of the talk will be based on joint work with Arpita Nayek. Here are the links of the preprints: \url{https://arxiv.org/abs/2207.01477} and \url{https://arxiv.org/abs/2302.00555}.