# Some easy singular Schubert varieties

I write $w$ for both an element of the Weyl group and the corresponding point in $G/B$. Let $X_w$ be a Schubert variety. Then $T_eX_w\subset T_e(G/B)$ is an inclusion of $\mathfrak{b}$ representations. The latter is generated by the $-\theta$-weight space, where $\theta$ is the highest root.

Suppose $w\geq s_\theta$, where $s_\theta$ is the reflection corresponding to $\theta$. Then the $\mathbb{P}^1$ connecting $e$ and $s_\theta$ lies in $X_w$ (think of the corresponding SL2), so the $-\theta$-weight space lies in $T_eX_w$. Since this generates $T_e(G/B)$, we have $T_eX_w=T_e(G/B)$.

Therefore if $w_0\neq w\geq s_\theta$, then $X_w$ is singular. This includes some of the first examples of singular Schubert varieties, for example the B2 singular Schubert variety and one of the A3 ones.