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?
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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 $k-1$ 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.
answered Nov 5, 2016 at 4:58
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$\begingroup$ You need to be sure that the curve is nondegenerate. This follows from e.g. Harris Algebraic Geometry Proposition 18.10. $\endgroup$ Commented Aug 11, 2021 at 10:22