# Motivation and checking for points being in “general position”

I wanted to know the motivation and the calculation details of checking that a certain number of points are in "general position".

Intuitively I was thinking that a set of points being in general position is a way of saying some sort of "non-degenerateness" as in they do not lie in some sub-projective space. But being in general position is defined in terms of saying that no subset (of maximum size of 2 more than the dimension of the projective space) of the points lies in some projective linear subspace. Hence I was wondering if any projective linear subspace is necessarily a sub-projective space?

To help fix ideas about how one goes about testing for being in general position or not let me consider the "textbook" example of the rational canonical curve from $\mathbb{CP}^1$ to $\mathbb{CP}^n$. Here one wants to say that the image of any set of points on $\mathbb{CP}^1$ is in general position. It is somehow related to the Vandder Monde determinant. I can't see how to set up this calculation.

By the definition one wants to check that there exists no linear transformation from $\mathbb{CP}^n$ to $\mathbb{C}$ such that the image points of any number of points under the above rational canonical curve is in its kernel. But I can't see how to go about this.

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I always thought "general position" was just a context-dependent license to exclude a measure-zero subset of the "problem space" from consideration ... –  Henning Makholm Sep 14 '11 at 17:55