Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

**185**

votes

**7**answers

90k views

### Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...

**131**

votes

**14**answers

9k views

### What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary ...

**126**

votes

**0**answers

6k views

### Grothendieck -sad news [closed]

Sorry for that this is not a real question. But I thought people would like to know.
Alexandre Grothendieck died today: ...

**124**

votes

**31**answers

16k views

### What should be learned in a first serious schemes course?

I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons ...

**107**

votes

**3**answers

5k views

### Analytic tools in algebraic geometry

This is not a very precise question, but I hope it will get some good answers.
As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign ...

**97**

votes

**7**answers

9k views

### How to memorise (understand) Nakayama's lemma and its corollaries?

Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...

**97**

votes

**1**answer

9k views

### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

**95**

votes

**1**answer

4k views

### What is a Frobenioid?

Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...

**92**

votes

**33**answers

56k views

### Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc.
One suggestion ...

**78**

votes

**6**answers

8k views

### what mistakes did the Italian algebraic geometers actually make?

It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations ...

**73**

votes

**9**answers

6k views

### Why are flat morphisms “flat?”

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is ...

**65**

votes

**8**answers

8k views

### What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The ...

**64**

votes

**2**answers

4k views

### Riemann hypothesis via absolute geometry

Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over ...

**63**

votes

**7**answers

6k views

### What is the field with one element?

I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is ...

**61**

votes

**2**answers

4k views

### Is it known that the ring of periods is not a field?

I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...

**60**

votes

**5**answers

6k views

### how does one understand GRR? (Grothendieck Riemann Roch)

I tried to answer an earlier question as to uses of GRR, just from my reading, although i do not understand GRR. Today i tried to understand the possible idea behind GRR. After editing my answer ...

**59**

votes

**2**answers

12k views

### Why is the Hodge Conjecture so important?

The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm ...

**59**

votes

**13**answers

19k views

### A learning roadmap for algebraic geometry

Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be ...

**58**

votes

**1**answer

4k views

### Is there a “classical” proof of this $j$-value congruence?

Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve ...

**55**

votes

**8**answers

4k views

### equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...

**55**

votes

**2**answers

4k views

### Has the mathematical content of Grothendieck's “Récoltes et Semailles” been used?

This question is partly motivated by this one.
Motivation
Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance the answers to this question or ...

**54**

votes

**4**answers

6k views

### Etale cohomology — Why study it?

I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...

**53**

votes

**6**answers

4k views

### Is there an analogue of curvature in algebraic geometry?

I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...

**53**

votes

**4**answers

3k views

### Rigidity of the category of schemes

Call a category $C$ rigid if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example ...

**51**

votes

**9**answers

3k views

### Taking “Zooming in on a point of a graph” seriously

In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...

**51**

votes

**2**answers

2k views

### The topological analog of flatness?

Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module.
Briefly the question is: what is the topological analog of this?
Many ...

**50**

votes

**2**answers

3k views

### The inverse Galois problem and the Monster

I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...

**50**

votes

**1**answer

4k views

### Smooth proper scheme over Z

Does every smooth proper morphism $X \to \operatorname{Spec} \mathbf{Z}$ with $X$ nonempty have a section?
EDIT [Bjorn gave additional information in a comment below, which I am recopying here. -- ...

**50**

votes

**0**answers

3k views

### Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmuller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here ...

**49**

votes

**14**answers

5k views

### Facts from algebraic geometry that are useful to non-algebraic geometers

A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic ...

**48**

votes

**6**answers

3k views

### Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications

If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications. I've heard informal examples (often involving moduli spaces), but am not sure of the ...

**48**

votes

**5**answers

3k views

### Is there an intuitive reason for Zariski's main theorem?

Zariski's main theorem has many guises, and so I will give you the freedom to pick the one that you find to be most intuitive. For the sake of completeness, I will put here one version:
Zariski's ...

**48**

votes

**1**answer

3k views

### Derived Functors Versus Spectral Sequences

Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories.
Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious ...

**47**

votes

**20**answers

9k views

### Algebraic geometry examples

What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)?
I'd prefer examples that ...

**47**

votes

**3**answers

7k views

### What's the “Yoga of Motives”?

There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new ...

**47**

votes

**7**answers

4k views

### Euler-Maclaurin formula and Riemann-Roch

Let $Df$ denote the derivative of a function $f(x)$ and $\bigtriangledown f=f(x)-f(x-1)$ be the discrete derivative. Using the Taylor series expansion for $f(x-1)$, we easily get $\bigtriangledown = ...

**46**

votes

**28**answers

4k views

### Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...

**46**

votes

**11**answers

6k views

### Non-commutative algebraic geometry

Suppose I tried to take Hartshorne chapter II and re-do all of it with non-commutative rings rather than commutative rings. Is this possible? Which parts work in the non-commutative setting and which ...

**46**

votes

**2**answers

1k views

### Reasons to prefer one large prime over another to approximate characteristic zero

Background:
In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather ...

**45**

votes

**4**answers

2k views

### What algorithm in algebraic geometry should I work on implementing?

This summer my wife and one of my friends (who are both programmers and undergraduate math majors, but have not learned any algebraic geometry) want to learn some algebraic geometry from me, and I ...

**44**

votes

**1**answer

2k views

### Did Grothendieck have a plan for proving Riemann Existence algebraically?

A recent question reminded me of a question I've had in the back of my mind for a long time. It is said that Grothendieck wanted the center-piece of SGA1 to be a completely algebraic proof (without ...

**44**

votes

**1**answer

1k views

### Local structure of rational varieties

I've been asked this question by a colleague who's not an algebraic geometer; we both feel that the answer should be "no", but I don't have a clue how to prove it.
Here's the question:
let $X$ be a ...

**43**

votes

**2**answers

3k views

### Polynomials having a common root with their derivatives

Here is a question someone asked me a couple of years ago. I remember having spent a day or two thinking about it but did not manage to solve it. This may be an open problem, in which case I'd be ...

**43**

votes

**2**answers

2k views

### a categorical Nakayama lemma?

There are the following Nakayama style lemmata:
(the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m_1, \ldots, m_n$ generate $M$ ...

**43**

votes

**2**answers

2k views

### What are reasons to believe that e is not a period?

I hope this rather soft question is suitable for MO, otherwise please migrate it to MSE.
In their paper defining periods [1], Kontsevich and Zagier without further comment state that $e$ is ...

**42**

votes

**3**answers

2k views

### Is “semisimple” a dense condition among Lie algebras?

The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some ...

**42**

votes

**5**answers

3k views

### A bestiary of topologies on Sch

The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? ...

**42**

votes

**3**answers

2k views

### Why do Todd classes appear in Grothendieck-Riemann-Roch formula?

Suppose for some reason one would be expecting a formula of the kind
$$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$
valid in $H^*(Y)$ where
$f:X\to Y$ is a ...

**42**

votes

**4**answers

3k views

### Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...

**41**

votes

**8**answers

4k views

### Questions about analogy between Spec Z and 3-manifolds

I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...