How do I start learning algebraic geometry? Like, what the fuck is a sheaf...

math.purdue.edu/~dvb/preprints/algeom.pdf

I found this to help get my shit started, are these good things to start on?

Shafarevich - Basic Algebraic Geometry or Mumford - Red Book of Varieties and Schemes

The point might be a generic point

First off, stop. You shouldn't be doing anything with sheaves. Scheme theoretic algebraic geometry will be a massive waste of your time without an understanding of why we're doing what we're doing. You should learn the more classical formulation, and learn it well -- I recommend Shafarevich. Comfort with complex analysis and differential geometry will do wonders for your intuition, by the way. Once you feel comfortable with varieties over algebraic closed fields, throw Hartshorne in the garbage. That book has zero pedagogical worth without an expert around to guide you. I think Ravi really is the gold standard for learning schemes.

Grad student in AG by the way; happy to answer any questions.

>You shouldn't be doing anything with sheaves.

I disagree. You don't have to jump right into schemes, but getting comfortable with geometry via locally ringed spaces asap is great.

It makes the jump to schemes more naturally if you already familiar with varieties (and even manifolds) being defined as locally ringed spaces.

I was reading the pdf I linked in I can see that were going to build up from affine spaces to the Zariski topology. But Im having trouble with the affine spaces.

Affine Space over an alg. closed field k is...

As a set: k^n

As a topological space: k^n + zariski topology

As a variety: k^n + zariski topology + sheaf of regular functions
The Nullstellensatz tells us there is a correspondence between k^n and maximal ideals of k[x1,..,xn].

So we could also say the underlying set of affine space is mSpec(k[x1,..,xn])

so there are no holomorhisms in alg geo?

In the Zariski topology you use regular maps.

However if you have a smooth projective variety over C, you can take the "analytification" and get a complex manifold. i.e. Take the same underlying set and equip it with the analytic topology and a suitable sheaf. The sections of the sheaf are then holomorphic functions.

You also have another topology called the Etale topology you can equip schemes with, which is like the analytic topology but more general. However it isn't a topology in the traditional sense, instead of being defined on a set (topological space) it is defined on a category (Site).

very useful definition

Most books covering classical material do so by implicit utilizing sheaf ideas anyways. It's better to not get caught up in dealing with terminology and needless generally while you're still getting comfortable.

You study topological spaces in general before going into specific examples, so why not study (locally) ringed spaces in general before going into examples?

By the time you study general topological spaces, you are usually familiar with a number of examples (at least R^n with its norms, maybe a discussion of metric spaces with many more examples).
Easing into it with the classical setting (varieties over acf) with a modern formulation as Shafarevich or maybe Perrin do it seems like a very natural idea. Many "real" difficulties (by which I mean, difficulties that require actual, not just formal, work to solve) already arise at that stage (basically anything that has to do with dimension, smoothness or multiplicities) and can be worked on without getting bogged down by the formalism, which makes it easier to understand why things are defined the way they are in the modern framework .

I totally disagree, since we spend a lot of time with R^n beforehand. Personally, I learned a lot of basic topological ideas through the lens of metric spaces in analysis, and I thought that was beneficial in much the same way. Math isn't just formally following a bunch of definitions; you really need to understand what they're saying.

>without getting bogged down by the formalism

I don't see structure sheaves as doing that, I think they actually clear things up. For instance the definition of the Zariski Tangent Space seems more natural once you realize it is the same thing as in differential geometry, but with a sheaf of regular functions instead of a sheaf of smooth functions.

Here's a redpill

you stop being a brainlet when you stop asking for help and start beating your head against the concept, reading and rereading it over and over, until you get it

You don't need the word sheaf to see any of that.

No you don't. But it is natural if you do.

No, it doesn't even add anything. The manifold case is defined by germs of derivations. You can easily define this in terms of the local ring at a point.

underrated

For any locally ringed space [math]\left( {X,{\mathcal{O}_X}} \right)[/math], the Zariski tangent space at a point p is defined by [math]{T_p}X = \operatorname{Hom} \left( {{\mathfrak{m}_p}/{\mathfrak{m}_p}^2,\kappa \left( p \right)} \right)[/math] where [math]{\mathfrak{m}_p} \triangleleft {\mathcal{O}_{X,p}}[/math] is the maximal ideal of the local ring given as the stalk of the sheaf.

This is isomorphic to. [math]\operatorname{Der} \left( {{\mathcal{O}_{X,p}},\kappa \left( p \right)} \right)[/math], which, if X is a the top. space of a manifold and [math]{\mathcal{O}_X} = C_X^\infty [/math], is the usual definition.

That's exactly what I said, except now you've introduced the concept of stalk of a sheaf just to use the ring of functions regular at P. None of that machinery adds anything.

>just to use the ring of functions regular at P.

No I used it to show that in this formalism you can unify definitions for both varieties and manifolds. What you call germs at p in differential geometry, is the stalk of the sheaf of smooth functions. The "ring of functions regular at p" is the same thing, it is the stalk of the sheaf of regular functions at p.

Yes, I'm aware of this. Why do you keep saying the same thing over and over again? The same analogy holds without using the word sheaf anywhere.

But it is not a natural equivalence without the sheaf formalism.

Explain to me what's at all different from "thing works in differential geometry. Let's do same thing with varieties." That's how everyone came up with it anyways, and it's intuitive without being forced to learn a massive and unintuitive theory.