general point, general line - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T17:34:21Z http://mathoverflow.net/feeds/question/64089 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64089/general-point-general-line general point, general line John 2011-05-06T06:14:45Z 2011-05-09T22:19:52Z <p>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.</p> http://mathoverflow.net/questions/64089/general-point-general-line/64094#64094 Answer by Francesco Polizzi for general point, general line Francesco Polizzi 2011-05-06T07:52:20Z 2011-05-06T08:09:48Z <p>I hope that the following example, in the vein of Daniel's comment, can be useful for you.</p> <p>Let $H \subset \mathbb{P}^3$ be a fixed plane and let us consider the statement:</p> <p>"A general line $L \subset \mathbb{P}^3$ intersect $H$ in a single point".</p> <p>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 </p> <p>"a line corresponding to a point in the open dense subset $X \setminus \Pi_H \$ of $X$". </p> <p>Analogously, the lines containing a point $p \in \mathbb{P}^3$ form a $2$-plane $\Pi_p \subset X$. Then in the statement</p> <p>"A general line $L \subset \mathbb{P}^3$ does not contain the point $p$" </p> <p>the word "general" precisely means </p> <p>"a line corresponding to a point in the open dense subset $X \setminus \Pi_p \$ of $X$". </p> http://mathoverflow.net/questions/64089/general-point-general-line/64096#64096 Answer by Jesko Hüttenhain for general point, general line Jesko Hüttenhain 2011-05-06T08:19:12Z 2011-05-08T06:26:05Z <p>In short: The statement "Property $P$ holds for a general hyperplane/line/etc" should mean that property $P$ holds for <i>"almost every"</i> hyperplane, line, etc. Now the cruicial point here is how to formalize the notion of <b>almost every</b> hyperplane - you have to <b>model</b> 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 <b><i>open</i></b> (Edit 05/08) subspace of the hyperplanes (in this model).</p> <p>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".</p> <p>Now this might help understand S. Carnahan's comment: The <a href="http://en.wikipedia.org/wiki/Hilbert_scheme" rel="nofollow">Hilbert Scheme</a> is a construction that can serve as such a model for <i>all</i> closed subvarieties of a variety, hence allowing you to speak of a general hypersurface of degree $d$, for instance. </p>