Algebraic geometry

I'm an undergrad math student currently interested in algebraic topology and a bit of differential geometry. Lately, I've been curious about abstract harmonic analysis, Kolmogorov complexity and computer science related theories too, ofc category theory itself. I've noticed that many of my peers are heavily into algebraic geometry, even though the fundamental AG courses offered by my uni are at in grad level, they still have taken the time to self-study this field, also continuously advocating its central role in mathematics, suggesting me to start my AG journey as soon as possible. They often mention how it intersects with various branches of math, hinting at its fundamental importance.

Although I see algebraic geometry as a potentially valuable tool for future research, I'm still on the fence about diving into it. I guess I'm looking for more concrete motivations or applications that resonate with my current interests and can be grasped at the undergraduate level.

Could anyone share examples or fundamental reasons why studying AG might be beneficial for someone like me, whose primary focus has been on topology and analysis? I'm particularly interested in how AG concepts might connect to the areas I am studying or planning to explore.

I hear lots of hubbub about algebraic geometry being very interdisciplinary, but the results really don't seem like they're especially interdisciplinary. Rather, how can one tell if it's a very ID field, or if it's as ID as other fields but it has just been studied so extensively that results in many places come up?

It seems every other mathematician you meet these days has some involvement in algebraic geometry. I have also read that it is currently the most active area of research. As someone who is only a graduate student working in a field far away from algebraic geometry, why does anyone care about it? What makes it so interesting/popular? Does it solve classic open questions or rather create new ones? There has to be something that warrants the long list of prerequisites, but due to my ignorance I cannot see it. I have also read that it is motivated by, or at least applies to, physics (namely string theory). Can anyone elaborate on this?

In short, if you had to "sell" algebraic geometry to me, what would your pitch be?

I've been studying classical algebraic geometry, and am hoping to move on to the scheme theoretic formulation soon, but I've recently realized that I didn't actually know what algebraic geometry was "about." Wherever anyone describes algebraic geometry, it's as the study of "geometric" or "qualitative" properties about zero sets of polynomials. Also, whenever anyone asks about "the point" of modern algebraic geometry, the response consists of various arithmetic applications.

In the most general form, my question can be formed: what is goal of "pure" algebraic geometry (if that's even a thing)? In a more specific wording: what specific qualitative/geometric properties of varieties are algebraic geometers particularly worried about, and how does scheme theory help study those?

In the 70s, Mumford wrote "Algebraic geometry seems to have acquired the reputation of being esoreric, exclusive and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point of view is accurate."

Now I use algebraic geometry almost everyday in my research and I was wondering if this quote still holds some value today. So what is your opinion on algebraic geometry ? (And not algebraic geometers).

What are good introductory books on Algebraic Geometry for people with a background primarily in physics? Is it still necessary to learn about areas like commutative rings and homological algebra before starting AG, or are there books that teach the necessary ideas before jumping into AG?

First the background: A year ago I took an algebraic geometry course, and we covered sheaves, schemes (toric schemes), proj construction and that kind of stuff. I have only ever seen the variety perspective hand-waved around from the professor, but we never had a definition of those on the blackboard.

The constructions we made were all fascinating, interesting and fun, but I can't say that I learned anything about geometry, nor am I able to answer any geometric questions I couldn't answer before. One of the few things that seemed to be related to geometry were toric schemes/varieties (because one can actually draw them; a process which involved the geometric capabilities of a third grader). Whenever we'd argue "geometrical" we would draw some pictures and argue about a topological space. In that sense algebraic geometry feels more like "topological algebra". Very frustrating.

So my question is this: Are there (reasonable) geometric problems one can tackle with algebraic geometry or is my notion of geometry just way off?

Hey,so i am currently doing my Master's degree and trying to figure out which of the remaining courses i want to attend. I have taken mostly courses in the context of Algebraic Topology / Differentiable Manifolds / Geometric Group Theory etc. Algebraic Topology was my absolute favorite. I have to admit that i didn't really enjoy all the calculus that comes with differentiable Manifolds, but that's another story.

This summer-term i am going to take two advanced courses on Homological Algebra which already excites me, i am really looking forward to it.

Now the thing is, Algebraic Geometry is quite prominent on my university and every algebra-related course seems to emphasize how important certain aspects are in Algebraic Geometry etc. Especially the Homological Algebra courses are already promoted as being prerequisites for the Algebraic Geometry lectures.

I absolutely loved the homological algebra we did in algebraic topology and i always enjoyed category theory. However, whenever i tried looking up what the motivation of Algebraic Geometry is, i couldn't get excited whatsoever. I have also never been interested in algebraic number theory to be honest. I never really cared about polynomials, but i don't want to be ignorant towards it.

I am aware that i don't know what i don't know. So here's my question: Is there a chance i am actually going to like algebraic geometry under the assumption that i really like category theory and homological algebra? I don't want to regret not having taken it if i eventually figure out that i would have actually enjoyed it.

Thanks for any feedback!

The first few items concern classical algebraic geometry, which studies polynomials over the reals and complex numbers.

1. Classical algebraic geometry is the 'next step' after linear algebra. Linear algebra allows only addition and multiplication by scalars. If you allow multiplication of coordinates, you get polynomials in multiple variables, and that's exactly what classical algebraic geometry is. There, one studies 'varieties' which are like linear subspaces. Instead of being defined by a linear equation (like x+y+z=0), they are defined by algebraic equations (like y+x³ -z=0). Despite some hiccups, dimension is well-behaved, and each extra equation usually cuts down the dimension by 1.

2. A lot of very interesting theorems came out of algebraic geometry, especially when projective space is thrown in. Projective space adds points at infinity to Euclidean space. So, for instance, two parallel lines in the plane intersect at infinity. This gives you theorems like Bezout's theorem, which says that two varieties (i.e. zero sets of algebraic functions) that aren't degenerate (like sharing terms) will intersect in the projective complex numbers a number of times equal to the product of their degrees (so two quadric surfaces in projective C³ intersect 4 times).

3. Classical algebraic equations come up all the time. Matrix multiplication is algebraic, the determinant equation is an algebraic equation, etc.

4. Now we get to crazier things. When abstract algebra was being developed, mathematicians began to see connections between polynomial rings and algebraic equations. In particular, every variety corresponded to a 'prime ideal' in the complex numbers and points corresponded to maximal ideals. What does this mean? Look at the complex numbers C. Take any polynomial in C. By the fundamental theorem of algebra, it factors into linear polynomials (x-a)(x-b)…(x-c). So the only 'prime' polynomials are the linear ones (x-a). But these are in 1-1 correspondence with points. It gets more complicated in higher dimensions.

5. So now mathematicians knew that you could make prime ideals in a ring correspond to points in a space. So they took it and ran with it: what if you took 'any' ring and made a space whose points corresponded to prime ideals? This is the idea of a 'spectrum'. It turns out that commutative rings of 'finite dimension' have the most tractable spectra. These spaces have weird properties. For instance, the element 0 usually forms its own ideal, but corresponds to no classical point. It's an extra point that is somehow close to all other points, because every other ideal contains it. The integers have a countable number of points, one for 0 and one for each prime. Things get really, really weird if you look at the prime ideals of Z[X], the ring of integer polynomials.

6. Mathematicians quickly discovered that almost everything you did with polynomial rings had a geometric analogue. Quotienting by a prime ideal corresponded to restricting to a subspace. Inverting a prime ideal corresponded to taking the complement. Quotienting by only the linear terms of an ideal gave the tangent space. Tensor products gave 'fiber products'. The possibilities were endless!

7. It even extended to number theory, giving number theory a geometric setting. In many ways, number theory could be done by working with countable spaces like the integers in the same way that you work with uncountable spaces like the complex numbers. This explains many of the connections between finite fields and complex numbers, such as the Riemann hypothesis's connection to certain finite fields.

8. Now algebraic geometry is even being applied to physics in the form of string theory. String theory is deeply involved in algebraic geometry.

These are just a few reasons that algebraic geometry is interesting to me.

As far as I'm aware, algebraic geometry is a supposedly massive field, but i can never find any subfields of it. Could anyone explain what the subfields of algebraic geometry are to me, and (at an undergrad level?) explain to me what they're about?

I'm going to start studying algebraic geometry (from Hartshorne's book), with the hope of learning about schemes, sheaves, and their cohomology (chapters II and III). However, I'm not really sure what the "big theorems" on these are. For example, in algebraic number theory (not class field theory), one can divide a basic course into four fundamental concepts: unique factorization of ideals, ramification theory, finiteness of the class number, and the unit theorem. I'm aware these aren't literally the only ideas of algebraic number theory, but I've sort of been able to orient my studies around these three theorems, and gauge my progress based on my understanding of these four ideas. Are there analogues of these "main theorems" in algebraic geometry?

This year i take a semester course in algebraic geometry using hartshorne chapter 2 section 1-4,i wish to self study after that using either qing liu or gortz(for exemple i prefer the structural sheaf using basis over the one used in hartshorne ),i know that you should learn about ox-module ,coherent sheaves and cohomology,but the course is so big that a lot can be skipped in a first course.

i know that can someone give me advice about what should be learned or recommend me a course notes that guide me in course material,and thanks.

Hello, I'm a graduate student and I'm really curious about algebraic geometry. I'm familiar with Atiyah and McDonald's introduction to commutative algebra, but looking further, I don't know which should be the next step (book) or a good path for algebraic geometry from basic to "graduate" level and research. I will be thankful for your help :)

There's been a flurry of activity recently about using algebraic geometry and category theory to replace topology as the foundations for functional analysis. In particular, Peter Scholze and Dustin Clausen have developed a program of condensed mathematics to do this (and there was a post last week about formalizing one of the main results in Lean).

As someone who is not a functional analyst, but somewhat adjacent to it, I've been reading about this and wrote a blog post with some thoughts. In short, I'm excited to see what new ideas this has to offer but also skeptical that it can efficiently handle questions that are needed for PDE analysis, so I think it is probably better suited as a companion theory rather than a full replacement.

Please let me know if you have any thoughts/ comments.

Edit: Peter Scholze responded to this post. I've posted his comment as an addendum at the end of the post. It's very much worth reading.

Edit: I will add a bit of context. He said this after introducing DVRs using four equivalent definitions. Here is the complete quote:

We're making all these definitions to study Number Theory, but then, as you grow up and you study more and more mathematics, you realize that Number Theory and Algebraic Geometry are really the same thing.

I was just wondering why he believes this.

It feels like a lot of the work that young geometers are doing is at the intersection between geometry and combinatorics: eg. toric geometry, tropical geometry, enumerative geometry, hyperplane arrangements, flag varieties, quiver varieties, Young tableaux, combinatorial hodge theory, etc. For someone who knows AG at the level of Hartshorne but little to no combinatorics (maybe generating functions, permutations & combinations), what are is a good entry point to learn more about combinatorial AG? Should I develop a better foundation first in combinatorics such as from Stanley I,II?

I was put in a position of having been exposed to concepts of Topos theory through books like Topoi by Goldblatt, Elementary Categories, Elementary Toposes by McLarty, and Sets for Mathematicians by Lawvere long before I started to learn algebraic geometry so it's odd to me that a lot of books on Algebraic Geometry actively shy away from the concept. In the first section of Vakil's The Rising Sea, he says 'This I promise: if I use the word “topoi”, you can shoot me.' In the preface for Milne's Étale Cohomology, he says "Only enough foundational material is included to treat the étale site and similar sites, such as the flat and Zariski sites. In particular, the word topos does not occur."

So I ask, where can I find a good reference for toposes in their original context that isn't SGA4?

"The study of shapes, and how those shapes are modified by transformations" seems a reasonable definition of "geometry" to me. If you don't like this, that's fine! Please give the definition of geometry you prefer, and then answer the question in the title using that definition.

As any undergraduate knows, algebraic geometry introductions usually start by talking about zeroes of polynomials, which are shapes. But the impression that I get is that the subject diverges from that very quickly; the many times I have read about AlgGeo concepts, zeroes of polynomials are not directly talked about much. It looks to me as though AlgGeo is about zeroes of polynomials only in the same way that economics is about coins or physics is about planetary orbits; yes, an intro to the subject will mention them, but they disappear not long after, and only a few economists and physicists really invest themselves in studying those two things.

Anyway, Roger Penrose did his PhD in algebraic geometry, but his heart wasn't in it towards the end, he was well on the way to becoming a physicist. He said in an interview once that he felt Algebraic geometry was not really about geometry, instead it was about structures you find in algebra.

Roger Penrose is a geometer in the following very traditional sense: his lectures and books are full of pictures. This is not true of AlgGeo; you're lucky if your textbook has a picture per 50 pages. That compels me to think that he is correct. But is he wrong?

To people who'd say AlgGeo is about geometry, can I ask why pictures are so few and far between in discussions of it? I know it definitely can't be because AlgGeo is talking about high-dimensional spaces with strange metrics, because that is also a description of Penrose's work...?

Disclaimer: I know a fair amount of differential geometry. I know only the very very basics of algebraic geometry.

Let M be a smooth manifold, R = C^\infty(M). Some things I noticed:

1. The maximal ideals of R are in bijective correspondence with the points of M.

2. The "varieties" of R are closed sets in M; at the very least it should be "most" closed sets but I haven't spent enough time thinking about this.

3. I think this means that R is precisely the set of continuous functions on Spec(R). We can start to talk about an "M scheme" as a topological space locally modeled on M. I think this will just be a normal smooth manifold but I'm not sure.

4. M is determined up to diffeomorphism by R.

5. The localization of R at a maximal ideal/point x is equal to the ring of germs of functions at x.

6. The same can we said for module theoretical localization and R-modules of sections of vector bundles.

7. R is NOT Noetherian.

I'm sure all of these are well-established, and probably the reasons for many of the definitions/constructions in algebraic geometry. I imagine 7) is a big reason why this perspective is not nearly as useful as it is in algebraic geometry. But the first 6 are very intriguing to me. This leads me to a couple questions:

1. Are there any interesting results/perspectives that can be obtained by following this line of thought?

2. Are there any useful applications of nontrivial commutative ring theory, e.g. localization or primary decompositions or whatnot, to real differential geometry? (That don't go through, like, representation varieties or complex algebraic geometry or whatever.)

Algebraic geometry is closer to a cult than a math domain. To enter it you must have a doctor letter proving that you're insane, and accept Hartshorne as your new lord and savior. Finally, you must use your friend who likes analysis as a human sacrifice to the allmighty Grothendieck.