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Let $f\in \mathbb{C}[z_{0},\ldots,z_{n}]$ be an irreducible homogeneous polynomial and $X=V(f)\subset\mathbb{P}^{n}$ the projective hypesurface associated. I want to find (if any) explicit generators of $H^{0}(X,T_{X})$, at least their expression in the affine charts $U_{i}$. For example when $i=0$ i set $$x_{k}=\frac{z_{k}}{z_{0}}$$ and i'd like to find a tangent vector field to $X$ in the form $$\xi=\sum_{k=1}^{n}q_{k}(x_1,\ldots, x_{n})\partial_{k}$$ with $q_{k}$ rational functions. Is this problem solvable? Is there a computer program that can give me this result?

Thank you in advance.

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If $X$ is smooth, there is no non-trivial vector fields unless $X$ is a quadric or a plane cubic : see for instance… . – Olivier Benoist Jun 25 '13 at 16:05
I allow $X$ to be normal with isolated singular points – Italo Jun 25 '13 at 16:09

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