# Tagged Questions

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

**185**

votes

**7**answers

91k 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 ...

**97**

votes

**33**answers

57k 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 ...

**33**

votes

**6**answers

3k 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 ...

**34**

votes

**5**answers

3k views

### Is there a “geometric” intuition underlying the notion of normal varieties?

I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot.
One thing that strikes me ...

**26**

votes

**8**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 ...

**5**

votes

**2**answers

447 views

### Is $G/T$ a projective variety?

Let $G$ be a semisimple Lie group and $T$ be its maximal torus. Can we say that $G/T$ is a projective variety?. Is there any proof or counterexample for it?

**4**

votes

**1**answer

527 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 $\Delta : Y \to Y \times Y$ ...

**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 ...

**52**

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 ...

**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 ...

**35**

votes

**5**answers

5k 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 ...

**40**

votes

**1**answer

2k views

### What is the relationship between motivic cohomology and the theory of motives?

I will begin by giving a rough sketch of my understanding of motives.
In many expositions about motives (for example, www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined to be ...

**17**

votes

**5**answers

5k views

**26**

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)
...

**19**

votes

**6**answers

3k views

### Heuristic behind the Fourier-Mukai transform

What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?

**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 ...

**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 ...

**16**

votes

**5**answers

2k 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

**1**answer

564 views

### geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring.
What is the geometric interpretation of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric ...

**27**

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 ...

**21**

votes

**5**answers

1k views

### Verlinde's formula

"Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT.
Depending on...
• which chiral CFT one considers (does one restrict to WZW models, or not?)
...

**9**

votes

**4**answers

3k 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/
...

**8**

votes

**1**answer

2k views

### A nice explanation of what is a smooth (l-adic) sheaf?

I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.

**21**

votes

**2**answers

1k views

### What is the theory of local rings and local ring homomorphisms?

It is well-known that the category of local rings and ring homomorphisms admits an axiomatisation in coherent logic. Explicitly, it is the coherent theory over the signature $0, 1, -, +, \times$ with ...

**20**

votes

**1**answer

947 views

### Why is there a connection between enumerative geometry and nonlinear waves?

I'm not 100% sure that this question is appropriate for this site. If it's not, please tell me and I'll delete it.
Recently I encountered in a class the fact that there is a generating function of ...

**14**

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

**4**answers

2k views

### Isomorphism between varieties of char 0

Hi,
the following statement appeared implicitly in a text I read and maybe you could just
give me a hint how to see this resp. give a reference:
If you have two k-varieties $X$ and $Y$ (sufficiently ...

**29**

votes

**2**answers

1k views

### Is every Noetherian Commutative Ring a quotient of a Noetherian Domain?

This was an interesting question posed to me by a friend who is very interested in commutative algebra. It also has some nice geometric motivation.
The question is in two parts. The first, as stated ...

**9**

votes

**3**answers

2k views

### Existence of fine moduli space for curves and elliptic curves

For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...

**11**

votes

**1**answer

2k 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 ...

**17**

votes

**3**answers

1k views

### Fields of definition of a variety

Let K and L be two subfields of some field. If a variety is defined over both K and L, does it follow that the variety can be defined over their intersection?

**12**

votes

**1**answer

614 views

### Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$

The automorphism group of the algebra of $n$-dimensional matrices over a field $K$ is $PGL_n(K)$. The automorphism group of $n-1$-dimensional projective space over $K$ is also $PGL_n(K)$. Therefore, ...

**12**

votes

**2**answers

560 views

### Integral points on varieties

I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties.
For example, consider $X = \mathbb{A}_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine ...

**4**

votes

**2**answers

670 views

### Connections between Standard, Hodge and Tate conjectures on algebraic cycles?

What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?

**13**

votes

**2**answers

987 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 ...

**11**

votes

**3**answers

880 views

### What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?

More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...

**16**

votes

**1**answer

1k views

### What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...

**7**

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 $\phi: \pi_1(X \backslash \Delta) \to S_d$ ...

**4**

votes

**2**answers

487 views

### Vector Bundles on normal surfaces

Let $X$ be a projective normal surface over $\mathbb{C}$. In this related question it is stated as soon as $X$ is smooth any vector bundle defined on the compliment of a codimension 2 subset extends ...

**1**

vote

**4**answers

1k views

### A proof for a statement about polynomial automorphism

I already got a proof for the fact that if a polynomial map is surjective then it is also injective. However, I used the invariant dimension of a ring and I want a simpler proof. Bravo for any try. ...

**10**

votes

**0**answers

324 views

### What is the state of art in Groebner bases

How big polynomial systems can we deal with? How do you know when you don't even have to try?
Motivation:
Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...

**9**

votes

**1**answer

435 views

### Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...

**6**

votes

**3**answers

595 views

### Blow-ups and blow-downs

Can $\mathbb P \times \mathbb P \times \mathbb P$ be obtained from $\mathbb P^3$ by a finite succession of blow-ups and blow-downs along non-singular centers?

**6**

votes

**1**answer

610 views

### Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.
A fusion ring is given by a finite set of integer ...

**5**

votes

**1**answer

232 views

### Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...

**3**

votes

**2**answers

351 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 ...

**16**

votes

**1**answer

925 views

### For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?

My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...

**15**

votes

**1**answer

442 views

### Is the regularity of finitely generated rings decidable?

Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by regular that the localization at every prime ideal is a regular local ...