# Generic Points to the Italians

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!

• 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. Commented Nov 11, 2013 at 15:07
• 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. Commented Nov 11, 2013 at 15:27
• Most likely, a generic point was literally just an arbitrary point drawn from a dense open subset, or something thereabouts. Commented Nov 11, 2013 at 15:47
• 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". Commented Nov 11, 2013 at 16:40