Question

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.

Answer 1

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 $X$".

Answer 2

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.