# Blow up of a quadric in $\mathbb{P}^3$ in two points

I was reading in Griffiths and Harris about how the blow-up of a quadric at one point is isomorphic to a blow-up of $\mathbb{P}^2$ at two points, and how the two points on the $\mathbb{P}^2$ in this case correspond to the two lines on the quadric passing through the point being blown up. To understand this better I am trying to write some of this down in concrete equations, but I think I need some help. Here's what I've done so far:

Say we have a smooth quadric $Q$ in $\mathbb{P}^3$ given by equation $x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0$, and say we want to blow up at the point $(0:0:1:1)$ on the quadric. I'll look at this in the affine patch where we can take $x_4 = 1$. Then the blow up has coordinates $[(x_1,x_2,x_3);(a_1:a_2:a_3)] \in \mathbb{A}^3 \times\mathbb{P}^2$ sattisfying the equations $a_1x_2 = a_2x_1; a_1x_3 = a_3x_1; a_2x_3 = a_3x_2;$ and $x_1^2 + x_2^2 + x_3^2 - 1 = 0$. I tried from here to determine the exceptional divisor, the strict transform of $Q$ and the intersection of the exact coordinates of the intersection of the two. I tried fixing $a_1=1$ to work in affine coordinates and plugging into the equation of the quadric, but my attempts don't make sense to me. Could you help me with that?

These two points should then correspond to the two points on $\mathbb{P}^2$ that are being blown up and the exceptional divisor should correspond to a line through them according to G&H. Could this be made more explicit, as in writing down the equation of the line in $\mathbb{P}^2$?

Thank you.

• It may take some work to make this precise, but this is probably easiest to understand by thinking of the quadric and the plane as different compactifications of $\mathbb C^2$. The quadric compactifies via two lines meeting at a point (take this to be the point you are blowing up, so the two lines are determined). The plane compactifies by a single line (take this to be the line connecting the two points you are blowing up). In each case after blowing up, you have a compactification of $\mathbb C^2$ by a chain of three lines, each of which has self-intersection $-1$.... – Tabes Bridges Jun 2 '18 at 20:21
• .... and of course, blowing down the middle line in the chain recovers the quadric, while blowing down the outer two lines recovers the plane. If you know the classification of minimal rational surfaces (which I guess you may not at this point), you should be able to identify one surface from the other just by working out the full intersection theory. – Tabes Bridges Jun 2 '18 at 20:23
• If you know toric geometry, it becomes a very fun exercise ! – Nicolas Hemelsoet Jun 3 '18 at 10:30