Tangent cones to Schubert varieties in types $A_n$, $B_n$ and $C_n$

TL;DR

This paper investigates the geometric structure of tangent cones to Schubert varieties in types A, B, and C, proving their distinctness for different involutions in the Weyl group, with implications for algebraic geometry.

Contribution

It establishes the uniqueness of tangent cones and reduced tangent cones for Schubert varieties associated with different involutions in types A, B, and C.

Findings

Tangent cones are distinct for different involutions in types B and C.

Reduced tangent cones are distinct for different involutions in types A and C.

Results help distinguish Schubert varieties via their tangent cone structures.

Abstract

Let $G$ be a complex reductive group, $T$ be a maximal torus of $G$, $B$ be a Borel subgroup of $G$ containing $T$, $W$ be the Weyl group of $G$ with respect to $T$. To each element $w$ of $W$ one can associate the Schubert subvariety $X_w$ of the flag variety $G/B$, the tangent cone to $X_w$ at the identity point $p$ considered as a subcheme of the tangent space $T_p(G/B)$, and the reduced tangent cone to $X_w$ at $p$ considered as a subvariety of $T_p(G/B)$. Let $w_1$, $w_2$ be distinct involutions in $W$. We prove that if $G$ is of type $B_n$ or $C_n$, then the tangent cones corresponding to $w_1$ and $w_2$ are distinct. We also prove that if $G$ is of type $A_n$ or $C_n$, then the reduced tangent cones corresponding to $w_1$ and $w_2$ are distinct.