I was reading the General position article on Wikiepdia, and came across the following sentence:
General position is preserved under biregular maps – if image points satisfy a relation, then under a biregular map this relation may be pulled back to the original points.
Can someone point me to a reference where this is stated and proved? I cannot find any. I tried looking in Harris's Algebraic Geometry - A First Course and in Shafarevich's Basic Algebraic Geometry.
And just to make sure I understand, is this what the sentence from Wikipedia is stating?
Let $X \subset \mathbb R^m$ and $Y \subset \mathbb R^n$ be affine algebraic varieties and let $\phi : X \to Y$ be an isomorphism. Suppose $\{x_1, \ldots, x_k\} \subset X$ is such that for every $d \leq m+1$, no $d$ points of $\{x_i\}$ are not contained in an affine $(d-2)$-plane in $\mathbb R^m$. Then for every $d \leq n+1$, no $d$ points of $\{\phi(x_i)\}$ are not contained in an affine $(d-2)$-plane in $\mathbb R^n$.
(It seems like there's also a statement for projective varieties, but I think they are equivalent.)
EDIT: Thanks to Kenny Wong for pointing out that the statement above is not was is meant by the sentence from Wikipedia.
However, in this case, I am confused the argument that 5 points in general linear position determine a conic in $\mathbb R^2$. I thought this was the argument:
Start with 5 points $\{x_i\} \subset \mathbb R^2$ in genereal linear position. Let $y_i \in \mathbb R^5$ be the image of $x_i$ under the Veronese embedding. Since the Veronese embedding is an isomorphism onto its image, the points $\{y_i\}$ are in general linear position. Hence, there is a unique affine hyperplane passing through these 5 points, which gives us the unique conic.
In the sentence in bold, don't we use the fact that the Veronese embedding preserves general linear position? Is this something special for the Veronese embedding (i.e., is not a property of all biregular maps)?