Hello, could anyone explain the notion of ''general point'' and ''general line'', ''general hyperplane'' in algebraic geometry, What does it mean exactly general line in the 3 dimensional projective space? Thank you.
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1$\begingroup$ It is the tautological object over the generic point of the appropriate component of the Hilbert scheme. See en.wikipedia.org/wiki/Hilbert_scheme $\endgroup$– S. Carnahan ♦May 6, 2011 at 6:33
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11$\begingroup$ @S. Carnahan: Your answer may be over the OP's head. @John: This question probably isn't appropriate for this forum; you should maybe try math.stackexchange.com instead, or read Harris's Algebraic Geometry: A First Course. In short, when we say some property holds for a general line, point, etc. we mean that it holds for a Zariski-dense open in the space of lines, points, etc. $\endgroup$– Daniel LittMay 6, 2011 at 6:51
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In short: The statement "Property $P$ holds for a general hyperplane/line/etc" should mean that property $P$ holds for "almost every" hyperplane, line, etc. Now the cruicial point here is how to formalize the notion of almost every hyperplane - you have to model all hyperplanes: In other words, give them the structure of, say, a variety. Then you can say that some statement holds for almost every hyperplane if the property holds for a Zariski-dense open (Edit 05/08) subspace of the hyperplanes (in this model).
As Francesco already pointed out, this is exactly what the Grassmannian does for the linear subspaces of a vectorspace. It can be modelled as a projective variety via the Plücker embedding and then, it makes sense to say that some property holds for "almost every linear $n$-dimensional subspace".
Now this might help understand S. Carnahan's comment: The Hilbert Scheme is a construction that can serve as such a model for all closed subvarieties of a variety, hence allowing you to speak of a general hypersurface of degree $d$, for instance.
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3$\begingroup$ The three most usual ways to formalize what "almost every" means in Algebraic Geometry are: 1) For a Zariski dense open subset of the "modelling variety" (usually a "general" line means this). Daniel Litt's comment. 2) For the 'generic' point of the "modelling variety" (usually a "generic" line means this). The generic point exists and is dense if the variety is irreducible; it belongs to every Zariski open, which is then dense. S. Carnahan's comment. 3) For the intersection of countably many Zariski dense open subsets of the "modelling variety" (usually a "very general" line means this) $\endgroup$– quimMay 6, 2011 at 9:42
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$\begingroup$ @Jesko, quim is right, and I'm putting this comment here purely for emphasis. You generally want more than a Zariski-dense set. You want an open and Zariski dense set. $\endgroup$ May 6, 2011 at 14:06
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I hope that the following example, in the vein of Daniel's comment, can be useful for you.
Let $H \subset \mathbb{P}^3$ be a fixed plane and let us consider the statement:
"A general line $L \subset \mathbb{P}^3$ intersect $H$ in a single point".
One can think of it in the following way: the Grassmannian of lines $\mathbb{G}(1,3)$, via the Plucker embedding, can be identified with a quadric $X \subset \mathbb{P}^5$. The lines contained in $H$ give a $2$-plane $\Pi_H \subset X$. Therefore the word "general" in the statement precisely means
"a line corresponding to a point in the open dense subset $X \setminus \Pi_H \ $ of $X$".
Analogously, the lines containing a point $p \in \mathbb{P}^3$ form a $2$-plane $\Pi_p \subset X$. Then in the statement
"A general line $L \subset \mathbb{P}^3$ does not contain the point $p$"
the word "general" precisely means
"a line corresponding to a point in the open dense subset $X \setminus \Pi_p \ $ of $X$".
answered May 6, 2011 at 7:52