When I first learned algebraic geometry, I naturally wiki-ed the subject and there was a line there that said the old school Italians used the notion "generic points without any precise definition."

Now that I know what a generic point is, I am curious as to how did the Italians think about a generic point and/or what their intuition about it was and what kind of results did they prove using this notion.

Any answer would be greatly appreciated!

  • $\begingroup$ I think I remember seeing this discussed here before, and with some good references to the literature. Let me snoop around and see if I can find that. $\endgroup$ – hardmath Nov 11 '13 at 15:07
  • $\begingroup$ Likely Motivating Example for Algebraic Geometry/Scheme Theory is the discussion I was recalling. It has four highly upvoted Answers, plus links to other material, but the introduction of "generic point" by Italian geometers is not fully explained. $\endgroup$ – hardmath Nov 11 '13 at 15:27
  • 2
    $\begingroup$ Most likely, a generic point was literally just an arbitrary point drawn from a dense open subset, or something thereabouts. $\endgroup$ – Zhen Lin Nov 11 '13 at 15:47
  • $\begingroup$ In complex algebraic geometry, there is a notion of "k-generic point" for a subfield $k$ of $\mathbb C$. It is defined as a point $p \in V(I)$ such that every polynomial $f \in k[X]$ with $f(p)=0$ is in the ideal $I$, hence $f$ vanishes on all of $V(I)$. See the first pages of Mumford's book "Algebraic Geometry I; Complex Projective Varieties". $\endgroup$ – Fredrik Meyer Nov 11 '13 at 16:40

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