This is an idle question, but I give the example that motivated me below.

Say $X \subseteq {\mathbb A}^n_k$ is irreducible and $k$ is infinite. Then by picking a regular point of $X$ and picking equations from $X$'s ideal that cut out $T_x X$, we get a scheme containing $X$ as a component.

If we pick those equations generically, can we ensure that that scheme is a complete intersection with at most one extra component beyond $X$?

The example that got me wondering this is where $X = ${$(A,B) : AB = BA$} is the space of pairs of commuting matrices. Then one case of the above construction is $Y = ${$(A,B) : AB-BA$ is diagonal}, which is a reduced complete intersection with two components. I thought this was interesting but now I'm guessing it's the expected behavior.