Let $V$ be a vector space over $\mathbb{F}_p$ equipped with a symplectic form, which means that the dimension of $V$ equals $2m$. Consider the (proper smooth) variety $LG \subset GR(n,V)$ parametrising maximal isotropic subspaces $W \subset V$. I am interested in the following (closed) Deligne-Lusztig variety (where $()^p$ means Frobenius twist) \begin{align} S_V:=\{W \subset V \; | \; \operatorname{Dim} (W \cap W^{(p)}) \ge m-1 \}, \end{align} see for example (5.3) of https://arxiv.org/pdf/1301.1226.pdf and would like to compute its deformation theory. I know that this variety is irreducible and has dimension $m$ and moreover that it is smooth away from its $\mathbb{F}_p$ rational points. But I would like a canonical description of the tangent bundle over the smooth locus.
Standard deformation theory arguments will tells us that the tangent space of $GR(n,V)$ (the Grassmannian of $n$-dimensional subspaces $W$ of $V$) at a point $(W) \in GR(n,V)(k)$ is canonically isomorphism to $Hom(W, V/W)$. Now if $W$ in fact lies in the Lagrangian Grassmannian $LG$ then we get a canonical isomorphism $V/W \simeq V^{\ast}$ using the symplectic pairing and moreover \begin{align} Hom(W, V/W) \simeq Hom(W, W^{\ast}) \simeq W^{\ast} \otimes W^{\ast}. \end{align} One can show that the tangent space of $LG$ at $W$ is then given by $\operatorname{Sym}^2 W^{\ast} \subset W^{\ast} \otimes W^{\ast}$.
Question: Is there a similar description of the tangent bundle of $S_V$ over the smooth locus? If $W_0=W \cap W^{(p)}$ then the bundle $\hom(W/W_0, W)$ seems like a good candidate, but I haven't been able to prove this.
Bonus Question: If we blow up $S_V$ in its $\mathbb{F}_p$ rational points, is the resulting variety smooth?