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Let $X$ be an irreducible algebraic variety over some field $k$. It is well known, that if $X$ is smooth and of dimension $d$, then the tangent bundle of $X$ is smooth, irreducible and of dimension $2d$. There is no difficulty in defining the tangent bundle $T_X$ in the non-smooth case. It is defined as the (global) Spec of the symmetric algebra of the sheaf of Kähler differentials $\Omega^1_{X|k}$. In fact this definition works for an arbitrary morphism of schemes $X\to S$. (See EGA IV$_4$

All the irreducible components of $T_X$ come from points of $X$. See Prop. 2.2 in this paper for a precise statement. The generic point of $X$ always yields an irreducible component of $T_X$ of dimension $2d$, but typically there will be other components.

I would like to know if there is an example such that $T_X$ has an irreducible component of dimension strictly smaller than $2d$.

Note that this will not be possible if $X\subset\mathbb{A}^n_k$ is a complete intersection: If the ideal of $X$ is generated by $f_1,\ldots,f_{n-d}$ then $T_X$ is described by the $2(n-d)$ equations $f_1,\ldots,f_{n-d},d(f_1),\ldots,d(f_{n-d})$ in $\mathbb{A}^{2n}_k$. (Cf. Example 1.5 in the above paper.)

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up vote 3 down vote accepted

How about the union in $\mathbb{A}^9$ of the three $5$-planes, $\Lambda_1 = \text{span}(\mathbf{e}_1,\mathbf{e}_6,\mathbf{e}_7,\mathbf{e}_8,\mathbf{e}_9)$, $\Lambda_2=\text{span}(\mathbf{e}_2,\mathbf{e}_4,\mathbf{e}_5,\mathbf{e}_8,\mathbf{e}_9)$ and $\Lambda_3=\text{span}(\mathbf{e}_3,\mathbf{e}_4,\mathbf{e}_5,\mathbf{e}_6,\mathbf{e}_7)$? The Zariski tangent space at the origin is the entire $9$-dimensional affine space. Each pair $\Lambda_i$, $\Lambda_j$ is contained in a hyperplane $H_{i,j} = Z(x_k)$, where $k$ is the missing index. So a tangent vector at the origin contained in none of the three hyperplanes $H_{1,2}=Z(x_3)$, $H_{1,3}=Z(x_2)$, $H_{2,3}=Z(x_1)$ is not a "limit" of nearby tangent vectors, i.e., the Zariski tangent space of the origin is an irreducible component.

Of course this algebraic set is reducible. But since this problem is a formal local problem, I can make a "global" irreducible variety which is formally locally isomorphic to the algebraic set above (i.e., it is not "unibranch").

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