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Let $X \subset \mathbb{A}^{n}$ be an affine variety then we know that the dimension of $T_{p}(X)$ the tangent space of $X$ at a point $p$ is equal to $n$ minus the rank of the Jacobian (at p)

Is this true if $X$ is a projective variety? or at least $X=Z(f)$ where $f$ is a homogeneous polynomial? In case this is not true, what is the method to compute the dimension of the tangent space of a projective variety?

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up vote 4 down vote accepted

Let $X \subseteq \mathbb{P}^n$ be a projective variety defined by homogeneous polynomials $F_1, \dots, F_r \in k[x_0, \dots, x_n]$. Suppose you want to compute the dimension of the tangent space $T_pX$ of $X$ at the point $p \in X$.

Choose an affine chart $U_i = \{x_i \neq 0 \}$ of $\mathbb{P}^n$ that contains $p$, then the variety $X \cap U_i$ is affine (because is closed in $U_i \simeq \mathbb{A}^n$) and is an open subset of $X$, hence $T_pX = T_p(X \cap U_i)$. Therefore you can compute the tangent space of the affine variety $X \cap U_i$. Do you know affine equations of $X \cap U_i$? (Hint: dehomogeneize with respect to $x_i$!) Then apply the jacobian criterion.

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