# Tagged Questions

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

**29**

votes

**4**answers

2k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**72**

votes

**32**answers

43k 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 ...

**31**

votes

**5**answers

4k views

### Why tropical geometry?

Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being ...

**24**

votes

**2**answers

2k views

### How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)

What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me)
...

**39**

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 ...

**9**

votes

**4**answers

2k views

### One point in the post of Terence Tao on Ax-Grothendieck theorem

I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem.
http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/
...

**13**

votes

**3**answers

1k views

### When are (finite) simplicial complexes (smooth) manifolds?

Hi,
is there an algorithm that determines if a given simplicial complex is
a.) a manifold
b.) a smooth manifold
c.) homotopy equivalent to a manifold
d.) a real algebraic variety
?

**10**

votes

**1**answer

1k views

### What are the monomorphisms in the category of schemes?

Someone recently asked what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am ...

**12**

votes

**2**answers

836 views

### D-modules and Algebraic Solutions of PDEs

I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When ...

**6**

votes

**1**answer

1k views

### Which Riemann surfaces arise from the Riemann existence theorem?

The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi_1(X \backslash \Delta) \to S_d$ ...

**3**

votes

**2**answers

335 views

### Hopf algebra of Chevalley group from the root system

Has anyone worked out a uniform way of constructing the Hopf algebra of a Chevalley group out of the root system (or, more precisely out of the root datum for reductive groups).
By "uniform", I mean ...

**3**

votes

**4**answers

1k views

### Measure on real Grassmannians

OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix ...

**6**

votes

**2**answers

387 views

### Equivariant Stratifications of a Variety

Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are ...

**4**

votes

**1**answer

469 views

### Regular monomorphisms of schemes

In the category of schemes, the equalizer of two morphisms $f,g : X \to Y$ is always a locally closed immersion into $X$ (since this is just $X \times_{Y \times Y} Y$ and $\Delta : Y \to Y \times Y$ ...

**6**

votes

**2**answers

371 views

### An integrality question about expressing an integer as a product of numbers below $n$

Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as
$$
N= \prod_{j=1}^{n} j^{x_j}
$$
where $x_1$, $\ldots$, ...

**5**

votes

**1**answer

182 views

### Decomposition theorem for semi-abelian varieties

Fact :
Let $B$ an abelian subvariety of an abelian variety over a field $K$. We know that there exist an abelian subvariety $C$ of $A$ such that the restriction of addition gives an isogeny ...

**2**

votes

**1**answer

270 views

### Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they ...

**1**

vote

**0**answers

127 views

### Descending chain condition for radical ideals

For which integral domains $R$ (not filed) the ring $R[x_1, \ldots, x_n]$ satisfies descending chain condition for radical ideals? I am not expert in Ring Theory and I need an answer to construct some ...

**159**

votes

**7**answers

78k 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 ...

**118**

votes

**31**answers

14k 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 ...

**45**

votes

**20**answers

8k 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 ...

**93**

votes

**1**answer

8k 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 ...

**33**

votes

**14**answers

9k views

### The importance of EGA and SGA for “students of today”

That fact that EGA and SGA have played mayor roles is uncontroversial. But they contain many volumes/chapters and going through them would take a lot of time, especially if you do not speak French.
...

**33**

votes

**8**answers

5k views

### Sheaf cohomology and injective resolutions

In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...

**39**

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 ...

**61**

votes

**6**answers

6k 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 ...

**15**

votes

**17**answers

7k views

### Learning about Lie groups

Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.

**23**

votes

**6**answers

2k views

### “Points” in algebraic geometry: Why shift from m-Spec to Spec?

Why were algebraic geometers in the 19th Century thinking of
m-Spec as the set of points of an affine variety associated to the
ring whereas, sometime in the middle of the 20 Century, people started ...

**30**

votes

**9**answers

4k views

### What are dessins d'enfants?

There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led ...

**28**

votes

**5**answers

2k views

### Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation
I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...

**37**

votes

**2**answers

3k views

### What is the insight of Quillen's proof that all projective modules over a polynomial ring are free?

One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in ...

**29**

votes

**6**answers

2k views

### Why do Littlewood-Richardson coefficients describe the cohomology of the Grassmannian?

I'm looking for a "conceptual" explanation to the question in the title. The standard proofs that I've seen go as follows: use the Schubert cell decomposition to get a basis for cohomology and show ...

**21**

votes

**5**answers

4k views

### Why no abelian varieties over Z?

Motivation
I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form
the set ...

**31**

votes

**4**answers

2k views

### Motivation for concepts in Algebraic Geometry

I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns.
**
Question
**
Are there any well-motivated introductions to ...

**23**

votes

**7**answers

3k views

### Ubiquity of the push-pull formula

The push-pull formula appears in several different incarnations. There are, at least, the following:
1) If $f \colon X \to Y$ is a continous map, then for sheaves $\mathcal{F}$ on $X$ and ...

**20**

votes

**4**answers

2k views

### Using linear algebra to classify vector bundles over P^1

There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let ...

**35**

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 ...

**20**

votes

**3**answers

4k views

### When is the product of two ideals equal to their intersection?

Consider a ring $A$ and an affine scheme $X=SpecA$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds to the union ...

**52**

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 ...

**28**

votes

**3**answers

3k views

### Does “finitely presented” mean “always finitely presented”? (Answered: Yes!)

Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I ...

**12**

votes

**5**answers

3k views

### Nonsingular/Normal Schemes

I always had trouble remembering this. Is it true that a curve over a non-algebraically-closed field is normal implies that it's non-singular? How about a 1 dimensional scheme? How about dimension 2? ...

**28**

votes

**1**answer

681 views

### What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ ...

**28**

votes

**4**answers

2k views

### What motivates modern algebraic geometry for a combinatorial/constructive algebraist?

This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if not the question, ...

**27**

votes

**4**answers

1k views

### Does a scheme have a “separification”?

Background:
(1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful ...

**22**

votes

**7**answers

2k views

### When are there enough projective sheaves on a space X?

This question is being asked on behalf of a colleague of mine.
Let X be a topological space. It is well known that the abelian category of sheaves on X has enough injectives: that is, every sheaf ...

**18**

votes

**3**answers

2k views

### Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?

If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...

**16**

votes

**5**answers

1k views

### Why are finiteness conditions important (and how to recognize them)?

I think everybody here has met lots of finiteness conditions, like those requiring a vector space to be finite dimensional, an abelian group to be finitely generated, a ring to be Noetherian, a ...

**7**

votes

**4**answers

749 views

### Coboundaries and Gluing in Cech Cohomology - Intuition?

I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...

**26**

votes

**3**answers

2k views

### Wanted: example of a non-algebraic singularity

Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the singularity type of $p\in Spec(R)$ to be the isomorphism class of the ...

**26**

votes

**2**answers

2k views

### What interesting/nontrivial results in Algebraic geometry require the existence of universes?

Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that ...