Consider the quadric surface $X = \{ xy = zw \} \subset \mathbb{P}^3$ and pick a point $x \in X$. I think it is true that if we think of $\mathbb{P}^2$ as the space of lines through $x$ in $\mathbb{P}^3$, then the morphism $X \setminus \{ x \} \to \mathbb{P}^2$ which sends $y \mapsto \overline{xy}$ represents a birational map $X \to \mathbb{P}^2$. But I do not understand the geometry of $X$ well enough to prove this. Certainly this morphism fails to be injective along the two obvious lines in $X$ through $x$, but how do I see that the map is an isomorphism elsewhere? I would like to avoid computing in coordinates if at all possible.
projection of a quadric surface
-
0Nice question:+1 – 2012-05-14
2 Answers
The basic idea is that the "inverse map" is given by sending $\ell \in \mathbb{P}^2$ (identify the points in $\mathbb{P}^2$ with lines through $x$) to the point $y$ where $X \cap \ell = \{x, y\}$. (We are using that $X$ has degree 2, which means that it intersects a general line in 2 points.) It's pretty clear that this map is inverse to the one you described, wherever things are well-defined. It also should be clear that the sets of points in $X$ and $\mathbb{P}^2$ where the maps are not well-defined are proper closed subsets. Lastly, you need to check that these are actually morphisms where they are defined, and I'm afraid this step, by definition, requires a certain amount of coordinate computation to verify.
A side remark to help you understand the geometry of $X$ is that you can view it as the isomorphic image of $\mathbb{P}^1 \times \mathbb{P}^1$ under the Segre map $((a:b),(c:d))\mapsto (ac:bd:ad:bc)$ to $\mathbb{P}^3$. I chose a weird ordering for the products of the variables so that the equation $xy = zw$ would be satisfied, assuming you order your coordinates on $\mathbb{P}^3$ "alphabetically" as $(x:y:z:w)$. In particular, this shows that through every point of $X$ are exactly 2 lines in $\mathbb{P}^3$ contained in $X$. Indeed, the two copies of $\mathbb{P}^1$ give two separate rulings on $X$. Shafarevich's book has a nice discussion of this surface; you might also look at Igor Dolgachev's notes on Classical Algebraic Geometry.
-
0@JustinCampbell: To clarify, the precomposition with Segre means that $x(u,v) = a(u,v) c(u,v),$ $y(u,v) = b(u,v) d(u,v),$ etc. Diagramatically, this is just writing down the condition that the inclusion $\mathbb{P}^1 \hookrightarrow \mathbb{P}^3$ of the line into $\mathbb{P}^3$ factors through the embedding $X \hookrightarrow \mathbb{P}^3$ of your quadric surface $X$ in $\mathbb{P}^3$. – 2012-05-15
As a concrete example of this problem, consider the projection of $X$ through the point $[0:0:0:1]$ onto the plane $w=0$. Show this is a birational map (consider the Segre embedding above, where is it not defined?) and agrees with your construction above. I know this uses coordinates, but it is actually not too messy.
Edit: Let me elaborate a little bit. If you consider the affine patch where $w=1$ we get the equation $z=xy.$ The point mentioned above maps to $(0,0,0).$ This only contains the two families of lines mentioned in the previous post, so away from these two lines at the origin (namely the $x,y$ axes), the map to $\mathbb{P}^2$ is an isomorpism. Namely, send a point in $X$ to the the approriate line, and send a line to the associate intersection with $X.$
This picture helped me: http://bit.ly/KHkhme