I.4.3b

5/29/2021

This exercise is... weirdly put. First, look at the definition of a rational map:

Hmmmm, in the case that the image is k = A1, doesn't this just conicide with the definition of a function field? Indeed. And didn't we already handle that case in part (a)? Indeed. The only difference is that we now embed A1 into P1. But that shouldn't change anything about where the map f = x1∕x0 is defined, as f would still only be defined where x00, right, so isn't the answer the exact same as part (a)? Fucking nope, lol.

You see, you might get starting a bit suspicious when you suspect that the only thing we did is rename stuff, and the answer otherwise matches part (a), so you go ahead and look up the solution, and you see this:

 ψ : P2 -{(0, 0, 1)} → P1 (x0,x1,x2) → (x0,x1)

B-b-b-b-b-b-b-b-b-b-b-but how is that possible? T-t-t-t-t-t-t-the map f, which is what this part (B) is all about is given as f = x1∕x0. FOR INSTANCE: The point P = (0, 1, 1) is in the domain, and if you follow the prescription of f, it tells us to do 10 which is, obviously, not correct. And yet the solution includes P, and furthermore tells us simply that the image is actually (0, 1). WHY? HOW? WHAT THE FUCKING FUCK? WHERE THE FUCKING SHITFUCKING ANUSFUCK DID THAT COME FROM? STRAIGHT OUT OF YOUR ASS? EVERY SOLUTION ONLINE JUST MAGICALLY PULLS THIS ψ OUT OF NOWHERE. Let me tell you, reader: I got so fucking mad at this exercise. I was sitting in my chair just fucking fuming, but in order to avoid making noise and letting the other members of this household confirm their suspicions of me being a psychopath, instead of breaking something or yelling, all I did was kick off my house slippers. One hit the ceiling, the other hit the closet door. That was the physical embodiment of my rage: a pathetic, girly kick, like a teenage girl throwing a temper tantrum.

You have to interpret this question in a screwy way to get this to work. Recall from part (a) how I discovered the map for U0 A1 to be represented by the regular function

 (x0,x1,x2) x1∕x0

and then we are embedding this is P1:

 x1∕x0 (1,x1∕x0)

So on U0, you can write the composed map as

 (x0,x1,x2) (1,x1∕x0)

And now, since we're in projective coordinates, we can rewrite this as

 (x0,x1,x2) (x0,x1)

And taking this definition in and of itself, this is, indeed, the map brought up in the solutions, and one can easily see that it's indeed defined everywhere except (0, 0, 1).

So we're restricting to an open set, dividing by x0, embedding, and then "zooming back out" and then asking where is it defined. WTF.

One of the points of this exercise is to loosen up. This section in general is I think an opportunity to not be overprecise with definitions and stuff, because they're going to be very cumbersome for some of the stuff that goes on in this section. Neither f nor ϕ are actual set functions, and yet we give formulas for them. This is some abuse of notation that I wasn't expecting, but now I "get" it. Relax your muscles, lay back, stretch your slipperless feet, let your boner protrude from your sweatpants.