I'm sitting here. Here, at the cusp of a weekend full of "let's creepily lurk on random fediverse profiles." I know one person is having trouble with a particular piece of software, another is ticked off by a particular buzzword, another is having issues with their parents. A graph emerges: who knows who, who has had a feud with who, who is politically aligned with who, who is flirting with who. It's all there. It's all in the cloud baby *puts on stalker sunglasses*. Listen: You wanna know the network like the back of your hand? You want Memento style notes and maps splattered across your NEET walls? You wanna be able to do a fucking play-by-play of the Oregairu season 2 finale? If you want to find the truth, pay attention. Pay attention to the details. It's all in the facts. Everything is in the facts. The facts, the facts, the facts, the facts. Don't you see, God damn it!? *Vigorously grabs your shoulders* You just have to connect the dots. *Violently shakes you, saliva falling on your face* It's there, staring right at you God damn it, you bastard! You fucking bastard!..... *Comes back to my senses, releases your shoulders, takes a deep breath, faces away from you* The truth is hidden between the lines. All you have to do is look. *Leaves the room. Camera pans to you, profiling a troubled facial expression for 10 seconds before the scene fades out*

*Fade in. A delightful patio on a sprightly spring day*

"Well, hawt dawg, reader. Ain't we got a hawt dawgin harrrrrrt shorrrrnin' exshermasize 'ere in today's day, Yeehaw!"
"Ahaha, well, I suppose you're right, Mr. holeinmyheart. Ah, would you like some more tea?"
"Wellllll, don't mind if I hawt diggity dawgin do, readawg."
"Ahaha."
*You pour some more tea, and hand it to me, then continue,*
"'Readawg'. Now, that is certainly something I wouldn't have been able to come up with, Mr. holeinmyheart. You are such a funny, creative person. Sometimes I honestly just wish I could be as clever, and smart, and... handsome as you."
"Well, hawt dawg, I ain't that good lookin' yaknow? But I waaas preeeetty popular with the ladies, back in my day."
"Oh, is that so?"
"Yesseroonie. Ya shoulda seen me at them dances. Grindin' them twerrrrkin' senia ladies back when I was just a wee ollll' freshman. I'll tell ya, were those the times. Har har har."
"Ahaha. I bet you were quite the popular one back then... I wish people liked me."
"Well, well, what's the matter youngin', underrating your darn old self like that?"
"Oh, no. It's just. I'm not that attractive. And I don't stand out much. And, I'm not that smart either. Sigh I guess I'm just... good for nothing."
"Well, well, now, now. Don't get all hard on yeself like that. Yer plenny attractive. And ye gawt a darn swingin' personality, I'd say!"
"Do you really mean that, Mr. holeinmyheart?"
"Sure as my boots are strapped."
"Oh! You're just saying that to be nice!"
"C'mon, have a lil' more self confidance will ya?"
"But I'm ugly, and I'm dumb, and I'm lame, and, and, and...!"
"Listen, readawg. What do I gotta say to make ya darn believe me?"
"....W-well... There is one thing...."
"What is it, my darn diddly readeroni?"
"It's.... well....."
"Come on, don't be a darn diddly shyfly. Anything I can darn doodly do for ya, just ask."
"..."
You hesitate, then finally say, softly,
"W-will you kiss me?"
*Sip*


Okay, lol, that's enough. Of course it was going to end with me kissing (You) again. But anyway, I'm done with that for now. I darn diddly swe–I mean I swear. HERE'S THE EXERCISE AGAIN:



So I got stuck on part (a). In fact, I still can't solve it "directly". That's fine though, because part (b) is just a generalization of part (a). So if I just do part (b), which comes with a hint, part (a) is technically done!

...Part (b) still took me a while. Let me work through how I managed to get to the answer. We're studying the set Y ∩ H, right? Now, figuring out that it's nonempty is equivalent to figuring out that its complement is empty. So what I can actually study is:

(1)

There's other reasons why looking at it this way is helpful. Firstly, note that the hint refers back to 3.1e, which has something to do with singletons. Yes: If the above turns out to be a "mere singleton", then its likely that Y ∩H is nontrivial.

Additionally, note that we can actually rewrite the above as follows:

(2)

This suggests a connection to 3.5, which the hint also makes reference to.

So the proof is almost revealing itself: Show that the above is a singleton, by showing that it's both projective and affine. Something like that. Now, first of all: is (2) an affine variety? Well, according to the subspace topology of Pⁿ - H, Y is a closed set. So the isomorphism of 3.5 maps Y to an affine closed subset. Is it a variety (irreducible) though? Well, all we have to do is switch perspective: in the subspace topology of Y , Pⁿ -H is an open set. And looking all the way back at 1.6 tells us that is must be irreducible. So (2) is affine. (By the way, this is all assuming that (1)/(2) is nonempty, which is fine because if it were empty then we'd have Y ∩ H = Y , which would make it nonempty).

Okay, so it's affine. Yay! But remember: To apply 3.1e, I want it to be projective too. So how do I do this?

Actually, it doesn't have to be projective. Here's the trick: If Y ∩H isn't empty, then we have no work to do. But if it were empty, then we'd have that (1) is just Y , which is certainly a projective variety. But it's also affine, by the argument above. Which means that (1) would be a singleton, just like we wanted! And guess what? The only way for Y to be a singleton would be if dim Y < 1, which contradicts the exercise's assumption. Hence, Y ∩ H can't be nonempty. DARN DIDDLY DO–I MEAN DONE.