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.
closed as off topic by Ryan Budney, Dan Petersen, Daniel Litt, Sándor Kovács, Andy Putman May 10 '11 at 4:39Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question. 


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 Zariskidense 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. 


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 $X$". 

