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.
But then I realized that when you know so little, what can you talk about? Suppose you decide
to learn mathematics and make blog posts about it, who would want to read about you learning arithmetic?
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?
As someone on lainchan said: "Just do it. Don't think it in terms of content that need to be
produced. Just put whatever cross your head."
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 Neocitiesappropriate. 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 notsotrusty 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.
But then I realized that when you know so little, what can you talk about? Suppose you decide
to learn mathematics and make blog posts about it, who would want to read about you learning arithmetic?
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 peachcolored blog, which, as you say, doesn't get much viewership
because you don't have anything unique to say. "II 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, socalled "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 longwinded, 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 zeroset in
P^{n}, and
Z_{A} for the zeroset in
A^{n+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     

 = dim k[x_{0}…,x_{n}]∕I(Y )  1    

 = dim k[x_{0}…,x_{n}]∕I(Z_{P}(α))  1    

 = dim k[x_{0}…,x_{n}]∕
 1    

 = dim k[x_{0}…,x_{n}]∕I(Z_{A}(α))  1    

 = dim k[x_{0}…,x_{n}]∕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 ) = (f_{1},…,f_{q}) for some elements
f_{i}.
Hence
Y  = Z(I(Y ))  

 = Z(f_{1},…,f_{q})  

 = Z(f_{1}) ∩
∩ Z(f_{q})   
Ah, so
Y is the intersection of
q "surfaces".... But are they
hypersurfaces? Recall: according to
2.8, the
f_{i}s 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.
f_{1} is irreducible. Then
f_{1} = g ⋅ h for nonunits
g,h. Hence
 Z(f_{1}) ∩ Z(f_{2}) ∩
∩ Z(f_{q})  

 = Z(g ⋅ h) ∩ Z(f_{2}) ∩
∩ Z(f_{q})  

 = [Z(g) ∪ Z(h)] ∩ [Z(f_{2}) ∩
∩ Z(f_{q})]  

 = [Z(g) ∩ Z(f_{2}) ∩
∩ Z(f_{q})] ∪ [Z(h) ∩ Z(f_{2}) ∩
∩ Z(f_{q})]  

  
But since
Y is a variety (i.e. irreducible) either
Y = [Z(g) ∩ Z(f_{2}) ∩
∩ Z(f_{q})] or
Y = [Z(h) ∩ Z(f_{2}) ∩
∩ 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 f_{1} 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 f_{2},…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 P^{3} is a
settheoretic 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).