Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p_i\in X$. We may assume that $X$ has ordinary singularities at the $p_i$'s.
Does there exists a formula, perhaps in terms of $deg(X)$ of the multiplicities of $X$ at the $p_i$'s, for the dimension of $H^1(X,T_X)$, where $T_X = \mathcal{H}om_{\mathcal{O}_X}(\Omega_X,\mathcal{O}_X)$ ?
If not, does anyone know a good way to proceed in order to compute $h^1(X,T_X)$ ?
Put $d:=\deg(X)$. From the exact sequence $$0\rightarrow \mathcal{O}_X(-d)\rightarrow \Omega ^1_{\mathbb{P}^n|X}\rightarrow \Omega ^1_X\rightarrow 0$$you get an exact sequence $\ 0\rightarrow T_X\rightarrow T_{\mathbb{P}^n|X}\rightarrow \mathcal{O}_X(d)\rightarrow \mathcal{E}xt^1(\Omega ^1_X,\mathcal{O}_X)\rightarrow 0$.
From this you find easily $H^i(X,T_X)=0$ for $i\geq 2$. The sheaf $\mathcal{T}^1_X:=\mathcal{E}xt^1(\Omega ^1_X,\mathcal{O}_X)$ is a skyscraper sheaf concentrated at the $p_i$; its rank at $p_i$ is the Tyurina number $\tau _{p_i}$, which is easily computed in terms of a local equation for $X$. Taking Euler-Poincaré characteristics you get $$h^1(T_X)-h^0(T_X)= -\chi (T_{\mathbb{P}^n|X})+\chi (\mathcal{O}_X(d))-\sum \tau _{p_i}\ ,$$where each term on the right hand side is easily computable. If $X$ has no infinitesimal automorphisms, $h^0(T_X)=0$; I believe this is the case if $d\geq 3$ and $X$ is not a cone.
Edit : As pointed out by @Benjamin Tighe, my assertion that $H^{i}(X,T_X)=0$ for $i\geq 2$ is incorrect. Using $H^1(T_{\mathbb{P}^n|X})=0$, one should add to the right hand side of the formula $\dim \operatorname{Coker}\bigl( H^0(\mathscr{O}_{\mathbb{P}^n}(d))\rightarrow H^0(\mathcal{T}^1_X)\bigr)$. Even when the singular points are ordinary double points, this is delicate to compute: it is zero iff the singular points impose independent conditions on hypersurfaces of degree $d$.
