## Understanding the birationality of the projective plane and a nonsingular quadric surface

Today I’m going to write up a beautiful and very classical little piece of projective geometry, which is quite a departure from the sort of algebraic geometry I usually post about. Here is my abstract motivation: the projective plane and the surface both contain a copy of the affine plane (the former as a standard affine open and the latter as ). Thus these two varieties are birational, and it is well-known that a birational map of surfaces can be represented as a sequence of blow-ups and blow-downs. We will see, in a totally geometrical way, that is obtained by first blowing up two points of and then blowing down the line which connects them. Inversely, one could first blow up a point of and then blow down the two lines passing through that point to get .

Think of as the quadric surface via the Segre embedding, fix , and realize as the space of lines through in . Write for the two lines contained in , and consider the morphism defined by (by we mean the line in connecting and , or in our setup the projectivized tangent space ). It has an inverse given by the formula . Thus we have found a birational map which identifies copies of , and we claim this lifts to an isomorphism .

Now can be thought of as a subvariety of , where the first and second copies of consist of lines in through and respectively, or equivalently planes in containing and respectively. Namely, consists of triples satisfying . On the other hand, is the subvariety of consisting of pairs such that and if then is tangent to at .

First we exhibit a canonical isomorphism between the ambient spaces of the two blow-ups, so we must show how to identify with the space of pairs , where are planes with . For the quadratic curve contains the line , so it must be the union of two lines: write and . Now and must intersect in a point (they are members of the two distinct rulings of ), and in fact is the desired isomorphism. For the inverse, fix and write for the two lines which pass through . Then there are unique planes such that and , provided one orders and correctly.

To finish, let us show that the conditions which cut out the blow-ups coincide under our identifications of their ambient spaces. Let and write and , so if then this triple corresponds to . If it suffices to observe that . In the case we have , whence as needed. Conversely, given a pair , let be the corresponding triple, and note that when the lines and contain the same pair of distinct points and hence must be identical. If , then , which completes the argument.