Last exercise of this section! Y'know, something about forcing yourself to update a blog almost daily really makes you realize how hollow of a person you are. I wanted to do something "season finale"-ish for this season finale, but, as often is the case, when I sat before my text editor, I found myself stumped. Fiiiiiine~. Let's talk about being stumped.

One of my little hobbies is browsing Neocities sites by last updated and poking around in people's personal spaces, just like I do in real life (JUST KIDDING). And earlier today, I also had the idea of "hey, let's see what pages are tagged with math", and stumbled my way over to this blog post. A very relateable post indeed by an internet stranger struggling with lack of creativity as an adult.

ARRRrRGGGHHHH.... I'M MELTINGGGGG.... NOOOO ..... SAVe MEEEE.... HARTSHORNED HAS BEEN... INVALIDATED.. .... .... Yea. the author has a point. Which is why everytime I update this site, I also get this huge urge to delete everything. Every. Single. Time. It's actually reminiscent of the urge I get to erase myself when I say something stupid irl. I see a lot of articles nowadays, here on neocities included, criticizing the domination of social media and people's submission to consumption over production, etc. etc. I agree, I think, but I also have started to get second thoughts. When you get a boi like me in the creator's seat, are you sure you're doing the right thing?



This, honestly, is fantastic piece of advice... for people who are not me. No, reader, please, do your thing. I want to see you blossom and give birth (to content). It's just that.... errrrr..... listen. I have some... unpopular ideas. Like, very, unpopular. They are not Neocities-appropriate. We are all better off if I keep them out of my personal website and in my personal dungeon. Here's a piece of advice: If you're ever walking down the street and see a sign saying "JIGGLYPUFF SEX CENTER", don't enter.

AND YET: Here we are at the end of season 2. How is it, that your not-so-trusty admin has overcome the deletion urge after every single post? After every post, I am reminded of my hollowness, my lack of knowledge and utility. And yet, I am keeping the ball rolling daily...ish. It's simply the fact that, despite my epic sluggishness, I would probably have not made it this far in the book without the accountability of the blog. If you wanna force yourself to do something, make your life depend on it.



Let's RP. You're the sub and I'm the dom. You're a feminine young thang, learning arithmetic for the first time, and you decide to meekly post about it on your cute peach-colored blog, which, as you say, doesn't get much viewership because you don't have anything unique to say. "I-I just learned the commutative property today! That's so cool!" Hahaha, how cute.

Now enter me: The hung, sexy dom. I'm sort of a player, you could say. I get bitches and pump n dump them like they're nothing. By that I mean, I've dabbled in many hobbies and interests and have given up up them without acquiring anything beyond a passing familiarity with them. Actually, I'm quite pathetic. Actually, my penis is quite small.

Well, you see, my dabbling in a dearthless domain of dsubjects, filled to the brim with beginner tutorials and success stories of 0 to hero, had always left me wanting. Sure, the Medium blogger can detail his experience and his methodology in rising to his level of proficiency in a craft, but I'm not really feeling it, you know? Go online, get advice from random strangers? "You'll need a good teacher, and proper guidance, and..." blah blah blah blah blah SHUT UP. I'M A SCARY DOM. YOU CAN'T MAKE ME SUBMIT TO YOUR RESPONSIBLE WILES. Theoretically, people who got to where they were started just like me. But their stories just don't hit me here *points at chest*, you know? (might have something to do with the hole there)

But, perhaps, I stumbled on your useless, amateur, cute little arithmetic blog, and decided to follow along. And using that as a bit of encouragement, you continue onwards with your aimless adventure. And while you did, I see someone going through all the pain and gain that is involved with acquiring the knowledge of being able to add and subtract numbers with double digits. Instead of reading someone recount their experience, I watch you go through the little kinks, the long stretches of being stuck, getting lazy along the way, all the lamenting over the far road ahead. And I see your daily grind bring you all the way up to, say, a noncommutative group? Oh, how naive you were back in the day, my little sub. But now with the support of a single sexy follower, you've advanced. Or, rather, I've SPOILED you into a MATH WHORE. And now, after watching you, I'm truly convinced on the inspiring zero to buttslut hero story.

That is not a description of this blog (this blog is zero to negative one). But maybe it could be a description of yours, reader?

PART (a):
(Repasting for reference)


After getting stumped on an intro to this post, I got stumped on this exercise (BEHIND THE SCENES FOOTAGE: Did you know, the two famous, so-called "stump scenes" of 2.17, were actually shot in reverse order?). I was so stumped that I went all the way back to section 1, in a desperate hope for help. And then I rediscovered 1.9. Oh! This part is just a "projectivization" of 1.9, similar to the earlier exercises this section. Great, I just have to mimic/convert my 1.9 proof to do this.

....And then I realized that I was a bad writer in the early days of this blog, and my logic/reasoning is extremely difficult to make out. My head was spinning just trying to interpret my own nonsense there, and not to mention it's kind of a long-winded, complicated proof. I decided, "LET ME PROCRASTINATE AND LEAVE IT FOR TOMORROW".

But then: I thought of something very clever. I do not know how I thought of this, but I thought of it. Why don't I just use 1.9? Why don't I use the affine case to help me here in the projective case? Going back to my notation in 2.1, let write Z_P for the zero-set in Pⁿ, and Z_A for the zero-set in Aⁿ⁺¹. In which case, Y = Z_P(α) and let's set X = Z_A(α). In that case,

dim Y	= dim S(Y ) - 1	(thanks to 2.6)
	= dim k[x0…,xn]∕I(Y ) - 1
	= dim k[x0…,xn]∕I(ZP(α)) - 1
	= dim k[x0…,xn]∕ - 1
	= dim k[x0…,xn]∕I(ZA(α)) - 1
	= dim k[x0…,xn]∕I(X) - 1
	= dim X - 1
	≥ (n + 1 - q) - 1	(1.9)
	= n - q

Yup, I saved a lot of work by being clever. ONTO PART B

Part (b):

Repasting exercise for ref:



Lemme let q = n - r for simplicity. So I know that I(Y ) = (f₁,…,f_q) for some elements f_i. Hence

Y	= Z(I(Y ))
	= Z(f1,…,fq)
	= Z(f1) ∩ ∩ Z(fq)

Ah, so Y is the intersection of q "surfaces".... But are they hypersurfaces? Recall: according to 2.8, the f_is would have to be irreducible for me to conclude that they're hypersurfaces, which they might not be? What do?

Just break em up. Suppose, e.g. f₁ is irreducible. Then f₁ = g ⋅ h for nonunits g,h. Hence

	Z(f1) ∩ Z(f2) ∩ ∩ Z(fq)
	= Z(g ⋅ h) ∩ Z(f2) ∩ ∩ Z(fq)
	= [Z(g) ∪ Z(h)] ∩ [Z(f2) ∩ ∩ Z(fq)]
	= [Z(g) ∩ Z(f2) ∩ ∩ Z(fq)] ∪ [Z(h) ∩ Z(f2) ∩ ∩ Z(fq)]

But since Y is a variety (i.e. irreducible) either Y = [Z(g) ∩ Z(f₂) ∩ ∩ Z(f_q)] or Y = [Z(h) ∩ Z(f₂) ∩ ∩ Z(f_q)]. So we can get rid of one of them. Assume it's the former without loss of generality. Then

If g is reducible, just break it up again until it becomes irreducible (which must be possible since we're in a UFD. Or you can break f₁ up into its unique factorization to begin with and apply this logic). Hence, we can assume Z(g) is a hypersurface. And we just apply the same logic on f₂,…f_q and we're done.

Part (c)? NOT TODAY, MOTHERFUCKERS. I'm skipping starred exercises by default. But this is even worse: it directly contradicts my horrific 2.12d experience where I somehow proved that it can be generated by 2 elements. So, in preparation to at least attempt this exercise, I went back yet again to reread my 2.12d proof, and... I. STILL. CAN'T. FIND. THE. FUCKING. ERROR. FUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUCK. Yea, no thanks on this one.

And for obvious reasons, I'll have to skip part (d). But here are the references in case anyone is curious:




Those are some ollllld references. From what I could tell online, this problem is still unsolved. Though it's hard to find information online because how the fuck do you look up "closed irreducible curve in P³ is a set-theoretic intersection of two surfaces solved yet?" online. They should have given it a name, lol.

SEEYA IN SECTION 3 (it'll take a few days for me to read through it, so you can relax for a bit from my pink assault).