Reader: You may not know this, but I'm a hole of many talents. One of my many talents is doing impressions. And let's say, my skills are very refined. Yes: even more refined than all the brilliant talk show hosts that peddled Trump impressions for the past 4 years. Nay, nay, I don't mean to brag. In fact, if I may, if only for 10 seconds, kneel for the comedy genius, Stephen Colbert, and his mindblowing, breathtaking Trump impression. My. God. I just... I don't even know how to describe it. Every time Stephen Colbert, the comedy G-d, whipped out his Trump impression on the show, I was just... at a loss for words at the skill and artistry. It's so uncanny I sometimes even mistook it for Donald Trump himself. My only complaint is that he didn't use it enough. I know this is a hot take, but I honestly think over the past 4 years, Stephen Colbert should have utilized his Trump impression more often. That is merely my humble opinion. But I guess there is value in the ephemerality of great things, right? The sublime is often fleeting. Sometimes, something like Stephen Colbert's uncanny Trump impression may only appear once in a blue moon, if that blue moon perhaps shined every weeknight for 4 years straight multiple times in a single show. Actually, I'd like to take this opportunity to make a prayer. Guys: I know I'm usually a super funny, comical guy, but sometimes I have to get serious. Even the funniest men cry sometimes, just like Colbert himself did when Trump denied the election results. Now it's my turn to cry. I'd like to take a moment to cry, and pray. And I'd like you to join me. Indeed: I'd like to dedicate this prayer in praise of the YHWH of comedy, Stephen Colbert, and his sublime Trump accent. Let us have moment of silence for the next 20 dots.

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But anyway, my impressions are even better than the talk show kings of comedy. However, I have to preface this by saying that I have, let's say, a different style. And by that I mean, I do impressions in writing. Yes: I'm an expert at textual impressions. My voice irl is very sexy and dynamic, but alas, this blog is not voiced, and thus I must focus my efforts at mastery within the written form. But boy, do I have it mastered. I can do an impression like no other. And, here, in this post, I'll give you an exclusive example. I, holeinmyheart, present to you my impression of none other than... holeinmyheart. Yes: I will do an impression of myself over the course of section 4:



Thank you, thank you. *bows*. Ahahaha, yes, yes, it's quite a good impression. Told you. How did I get so good? I don't know. It must be my overflowing sense of empathy.

Indeed: The reason I'm skipping 4.8 and 4.9 is because I want to procrastinate on learning about field extensions as long as possible, so we've already arrived at the end of this section. LE SEASON BINALE =DDDDDDD

I basically already did this exercise back in 4.4b . Remember, letting t,u be coordinates for P¹ and letting t≠0 (i.e. t = 1), I got the union of two varieties in A³:

u	free
x	= 0
y	= 0

u	free
x	= u2
y	= u3

The former is the "exceptional curve" and the latter is "strict transform" I guess. They obviously only meet when u = 0, so at the point (0, 0, 0) in A³ or the point (0, 0, 1, 0) in A² × P¹. And this strict transform, as I said in 4.4b, is just the twisted cubic curve, which, again, is isomorphic to A¹.

Well, I'm pretty sure that's the intention here. The exercise is "done". But I still have a fuckload of questions.

First of all, to obtain the exceptional curve and strict transform, I'm only following the process outlined in example 4.9.1, but, uhhh.................. What the fuck is even going on.

WHAT THE FUCK IS GOING ON.

WHY IS THIS SO FUCKING UNCLEAR.

Look: First thing that's super sus (bro......... I can NEVER take breaking bad seriously after hank called walt a sussy baka.......... omg the internet is like totally INSANE. breaking bad? more like BREAKING MY MIND. i just cant believe that celebrities would cringily attach themselves to popular memes in 2021 this has NEVER HAPPENED BEFORE this is jsut IMPOSSIBLE guys) is that the example refers to the "strict transform" of the x axis and the "strict transform" of the y axis, whereas this exercise asks us to consider the "strict transform" of just Y as a whole. So.......... what the fucking fuck????? that wasn't defined, you sussy baka?????????

Now I resolved this by looking online and realizing that the "strict transform" is basically "the part of the inverse image of a variety that isn't the exceptional curve." So in the example, that Ỹ is the strict transform of Y , which confirms that my work is correct.

but this is STILL fucking confusing. Aren't we missing some points? Remember: We're only considering the part where t≠0 (t = 1). And yet this gives us the entirety of Ỹ = (ϕ^-1(Y - O))? What about the points corresponding to t = 0? Do they go in the exceptional curve or the strict transform? Because for the case of y² = x³ (i.e. this exercise) I admittedly get only one point: (0, 0, 0, 1). But where does this point go????? Is it not a part of Ỹ = (ϕ^-1(Y - O))? Why not? If instead of t≠0 we did u≠0, we appear to get something different. Is that also Ỹ = (ϕ^-1(Y - O)) somehow? And what about the point "left out" in that case?

I'm explaining my confusion very badly. my head is FUCKING BREAKING (BREAKIND BAD LOLOOLOLOLOLOLOOLOLOLOLOLOL). I'll try to think this over before moving onto the next section. I think it's valuable to spend at least some time "freely exploring" math instead of just trying to solve exercises. In any case, seeya in the next section