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$\_4$ 16.5.12.1.) 


All the irreducible components of $T\_X$ come from points of $X$. See Prop. 2.2 in this 
[paper](http://www2.im.uj.edu.pl/actamath/PDF/37-9-24.pdf)
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.)



Dimension of irreducible components of tangent bundle 
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"). 


