Suppose $X_{1}, X_{2} \subseteq \mathbb{A}^{n}$ are closed affine subvarieties such that $p:=(0,\dots,0) \in X_{1}\cap X_{2}$.
Define $X := X_{1} \cup X_{2}$.
I have shown the following inclusions for the tangent spaces:
$T_{p}X_{1} + T_{p}X_{2} \subseteq T_{p}X$.
However, I have been asked to give an example where the above inclusion is $\textbf{strict}$.
I have tried to mess around with a few examples that I can visualise (i.e. $n \leq 3$), but I can only seem to think of varieties that "intersect nicely". I have not been able to come up with an example, or indeed develop an intuition as to why the above inclusion can be strict.
Any help/hints would be appreciated!
Let $X_1 = V(y)$ and $X_2 = V(y-x^2)$.
Then the tangent spaces at zero of both these varieties is the $x$-axis.
But their union is the variety $W := V(y(y-x^2))$.
Then as the defining polynomial for $W$ has no linear terms, the tangent space at the identity is $\mathbb{A}^2$, which strictly contains the $x$-axis.
You can come up with this knowing that the intersection of any two irreducible components is a singular point, which, if both irreducible components are 1-dimensional, must have a 2-dimensional tangent space.