The uniform position theorem says that the points of a general hyperplane section of an irreducible curve lie in uniform position. Doesn't that imply that the points of a general subspace of codimension $k$ of an irreducible variety of dimension $k$ lie in uniform position?
1 Answer
Yes, of course. I assume that by a "general codimension $k$ linear subspace" you would mean something like the intersection of $k$ general hyperplanes.
A general codimension $k1$ linear subspace intersects an irreducible variety (I assume you include "reduced" in "variety") in an irreducible and reduced curve. Then the intersection of the original variety with a general codimension $k$ linear subspace is the same as the intersection of this curve with a general hyperplane, hence your desired statement follows from the original theorem you are mentioning.

$\begingroup$ You need to be sure that the curve is nondegenerate. This follows from e.g. Harris Algebraic Geometry Proposition 18.10. $\endgroup$ Aug 11, 2021 at 10:22