Algebraic Geometry Thread

did you mean affine schemes you dicklit?

What the fuck those that have to do with geometry

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>>8730156
Algebraic geometry is about the interplay between geometry and algebra. You can do geometry and prove stuff about algebra. And you can do algebra and prove stuff about geometry.

For example, a classic result is that every cubic surface has exactly 27 lines.

>pic related

>>8730170
>every cubic surface has exactly 27 lines.
What does that even mean

>>8730180
>>8730180
A surface is something described by polynomials in 3 variables. A cubic surface is something described by cubic polynomials in 3 variables.

What you might know are conic sections. These are described by quadratic polynomials in 2 variables, for example <math>x^2+y^2-1</math>
the points where this polynomial are zero is a circle.

>>8730180
look at the picture, there is a cubic surface, and you can see all the lines that it contains. There are 27 of them.

>>8730208
Lines where you fucktard? What kind of lines? Because I could draw any fucking number of lines I want.

>>8730212
straight lines unless you had others in mind?

>>8730216
So you mean entire straight lines contained in the surface.

Well, that's a clarification because whoever made that animation is a complete retard.

Try to look at a green line to try to see if looks straight. Because the fucktard decided to paint other lines green, it creates the illusion that the line kinda curves around the surface. We have like 50 quadrillion colors and the retard has to repeat them for his little animation. What a fucking retard. Algebraic Geometry confirmed for low IQ field.

>>8730315
Yeah so the claim is there are exactly 27 every time, no more, no less

How the FUCK do you even prove that in all cubic surfaces you have exactly 27 straight lines? In every single cubic surfaces?

>>8730199
><math>x^2+y^2-1</math>
[math]x^2+y^2-1[/math]
FTFY

>>8730330
By creating a theory that can categorize surfaces and then find a bunch of little details that must be true for all the members of that category of surfaces, and then use those little details to prove bigger details.

You know, like all of the mathematics. Read a book.

>>8730330
https://archive.org/stream/collectedmathema01cayluoft#page/444/mode/2up

>>8730340
Sorry im not an autistic faggot.

>>8730344
this is as old as egypt

>>8730340
there is a much simpler proof

>>8730344
Holy shit, that's a simple answer, thanks.

>>8730340
Go fuck yourself.

>>8730330
you prove that there are points on every cubic that have to have some lines go through them.

>>8730340
stop making shit up to confuse people faggot

>>8730142
>Every scheme is affine
>Algebraic Geometry is only about schemes

>>8730399
if yo scheme aint affine it sure is made up of some

>>8730315
Holy fuck what a tool.

>>8730408
Yes it is, but characterizing all of Algebraic Geometry by affine schemes is like characterizing all of Differential Geometry by R^n.

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fractals are cool, right?

>>8730315
Why are you acting like such a prig? OP has been very civil.

>>8730199
Thank you OP, very fascinating. What are the prerequisites for study in this field? Merely rudimentary algebra and geometry, or something more refined?

>>8730199
Why is it a circle?

>>8730142
hey, starting geometry course. The book is in the spirit of "erlangen programme"? what does this mean, is it an algebraic geometry course? Is this approach outdated?

>>8730170

holy fuck. thats neat anon.

I have no idea if this is relevant to algebraic geometry, but it seems that there is a connection between the number of ways to evenly divide a disc into n pieces with chords and the number of distance-regular simply connected graphs on n nodes. But I can't figure out what one has to do with the other.

>>8731221
The way to start is to do some commutative algebra. If you're into Geometry you won't enjoy this very much as it is really just pure algebra. Then you can look at classical algebraic geometry which should be enough to prove the 27 thing. The only problem however is this is pretty limited as you can't work in fields other than algebraically closed ones. For complex numbers its great other than that not so much. Modern algebraic geometry eliminates the need for fields to be algebraically closed by working over any field. Modern algebraic geometry uses schemes as the geometric object and commutative rings as the algebraic object. This provides a rich interplay between algebra and geometry which is why it has the name. The advantage of this approach is that it is applicable to Number Theory for example. Fermats Last Theorem is an application of scheme theory to classify and count elliptic curves and modular forms via the Modularity Theorem.

>modern algebraic geometry is the intersection of number theory, algebra and geometry

>>8731224
>erlangen programme
still very much group theory/lie theory. A geometric theory but not algebraic geometry. Algebraic geometry is a sophisticated theory in that it isn't just applied here and there willy nilly. Erlangen is pretty much doable with classical tools no need for modern ones.

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>>8732901
>Number theory grad student
>Prof says we'll need scheme theory to go further
>Proceeds to explain scheme theory for 2 hours
>Moves one to building modular forms on [math]\mathbb{Z}[\frac{1}{N}][/math]

I wasn't prepared for that.
Fucking schemes, how do they work ?

>>8733026
Read the first 2-3 chapters of Eisenbud&Harris.

>>8733038
Thanks, I'll check it out

>>8733026
If schemes were easy, everybody would use them. Schemes are like a really powerful sword. You have to train to use them. You can't get any old joe using schemes correctly.

>>8730475

Its worse, since smooth manifolds embed into some R^n. Proper varieties of positive dimension never embed into A^n, and starting in dimension 3 may not embed into P^n.

>>8733026
Try learning homotopy theory of schemes....