(cross-posted to MSE: https://math.stackexchange.com/questions/3884084/application-of-a-general-uniform-position-theorem)

A general version of the uniform position theorem on p. 113 - 114 of ACGH states the following:

Let $C \subset \mathbb{P}^r$, $r \ge 3$, be an irreducible, non-degenerate, possible singular, curve of degree $d$. If $\mathcal{D}$ is any linear system on $C$, and $\Gamma = H \cap C$ a hyperplane section general with respect to $\mathcal{D}$, then all subsets of $m$ points of $\Gamma$ impose the same number of conditions on $\mathcal{D}$. Equivalently, if $\Gamma' \subset \Gamma$ is any subset which fails to impose independent conditions on $\mathcal{D}$, then every divisor in $\mathcal{D}$ containing $\Gamma'$ contains $\Gamma$.

Just to clarify, does "general with respect to a linear system" mean that subsets of hyperplanes $(\mathbb{P}^r)^*$ imposing a smaller number of independent conditions form a strictly closed subset of the subset imposing $\ge m$ conditions? If so, does a similar statement apply to linear subspaces of complementary dimension for higher dimensional varieties? I tried to find a more precise statement in other sources, but most of the resources that I found tend to deal with hyperplanes whose intersection with $C$ are linearly independent and don't involve more precise information about the number of independent conditions.

EDIT: A link suggested while I was writing this question was the following: Will (general points + small number of arbitrary points) impose independent condtions on plane curves? The motivation for this question is essentially in the opposite direction in that we want to know something how "often" points of some variety fail to be in linear general position.



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