Local cohomology at generic singularities of Schubert varieties in cominuscule flag varieties

Abstract

We study the local cohomology of a Schubert variety in a co-minuscule flag variety at a generic singularity. Let $G$ be a reductive complex algebraic group with a Borel subgroup $B$, let $W$ be the Weyl group of $G$ with a set of coxeter generators $S$ and let $P$ be a maximal parabolic subgroup of $G$ corresponding to the set $S-{s}$, where $s$ is a co-minuscule node. Let $X_w$ be a Schubert variety in co-minuscule flag variety $G/P$ with a generic singularity $bar{v}$, where $v,w$ are minimal coset representatives in $W/W_{S-{s}}$ and $bar{v}$ is $vP/P$. We develop a process to compute the local cohomology of $X_w$ at $bar{v}$, $H^*(X_w, X_w-\bar{v})$, with respect to integer coefficients and examine whether or not $H^*(X_w, X_w-\bar{v})$ has torsion. We consider the Richardson variety $R_{vw}$, the intersection of $X_w$ with the opposite Schubert variety $X_v^{-}$. Further, we introduce the concept of emph{Thom variety}, a Schubert variety that is also the Thom space of a line bundle over a flag variety. Reformulating results due to Brion-Polo, we show that, barring the case when we consider a Schubert variety in co-minuscule flag variety of type $C_l/A_{l-1}$ with a generic singularity of a certain exceptional type, for each $R_{vw}$ there is a flag variety $M$ such that $R_{vw}$ can be identified with a Thom variety $T(\xi)$, where $xi$ is a special line bundle over $M$. The singularity $bar{v}$ of $R_{vw}$ corresponds to the point at infinity in $T(\xi)$. We prove a theorem that identifies all the possible types of flag varieties $M$ that correspond to $R_{vw}$ and also determine that $H^*(X_w, X_w-\bar{v})$ is just the suspension of $H^*(T(\xi), T(\xi)-\infty)$. We compute $H^*(T(\xi), T(\xi)-\infty)$ for each possible flag varieties $M$ using the emph{node-firing game}, a variant of Mozes' number-game. Using these computations we prove that $H^*(X_w, X_w-\bar{v})$ is torsion-free when flag variety $G/P$ is simply-laced and has 2-torsion when $G/P$ is of type $B_l/B_{l-1}$.