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 $h^1(X,T\_X)$ ?



Cohomology of tangent sheaf of a singular hypersurface
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 $d$.


