# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

1,261
questions

**40**

votes

**2**answers

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### 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 ...

**180**

votes

**35**answers

115k 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 ...

**55**

votes

**6**answers

6k views

### Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?

**31**

votes

**2**answers

7k views

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?
Please see this related post and also the following post.. For Mathematical ...

**32**

votes

**1**answer

3k views

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

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...

**272**

votes

**7**answers

127k 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 ...

**55**

votes

**5**answers

7k 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 ...

**14**

votes

**1**answer

454 views

### Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?

The q-Vandermonde identity reads:
$$ \binom{m + n}{k}_{\!\!q} =\sum_{j} \binom{m}{k - j}_{\!\!q} \binom{n}{j}_{\!\!q} q^{j(m-k+j)} $$
The q-binomial coefficients:
$$ \binom{ a }{ b}_{\!\!q} $$
...

**41**

votes

**8**answers

5k 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 ...

**23**

votes

**2**answers

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### When is fiber dimension upper semi-continuous?

Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$.
When is this function ...

**16**

votes

**4**answers

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### 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 ...

**16**

votes

**3**answers

3k views

### Extending vector bundles on a given open subscheme, reprise

In this question, Ariyan asks about the question of uniqueness of extensions of vector bundles when they exist.
Sasha's answer suggests that extensions of vector bundles don't always exist.
More ...

**16**

votes

**3**answers

1k views

### what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are ...

**36**

votes

**1**answer

1k views

### Is an affine fibration over an affine space necessarily trivial?

Let $X$ be an algebraic variety over an alg. closed field with zero char. and let $f:X\to \mathbb{A}^n$ be a smooth surjective morphism, such that all fibers (at closed points) are isomorphic to $\...

**6**

votes

**1**answer

1k views

### Bijection implies isomorphism for algebraic varieties

Let $f:X\to Y$ be a morphism of algebraic varieties over $\mathbb C$. Assume that
a) $f$ is bijective on $\mathbb C$-points
b) $X$ is connected
c) $Y$ is normal.
Does it imply that $f$ is an ...

**14**

votes

**2**answers

789 views

### Explicit invariant of tensors nonvanishing on the diagonal

The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...

**8**

votes

**1**answer

696 views

### When is the kernel of the etale fundamental group in a fibration abelian?

Let $X \to Y$ be a smooth proper morphism. Let $y$ be a geometric point of $Y$. Is the kernel of the natural map of etale fundamental groups $\pi_1^{et}(X_y) \to \pi_1^{et} (X)$ abelian?
This is true ...

**4**

votes

**2**answers

530 views

### Non-projective smooth complete threefolds with a pair of points intersecting every surface

I've been learning about non-projective complete varieties and I am trying to get a handle on how crazy they can get.
$\textbf{Question:}$ Let $V$ be a complete threefold over $\mathbb{C}$. Given ...

**2**

votes

**1**answer

299 views

### Bounding the number of critical points in a Lefschetz pencil

Let $k$ be an algebraically closed field. Let $X/k$ be a smooth projective variety. For a suitable embedding in $\mathbb{P}^{n}$ we can form a Lefschetz pencil $\widetilde{X} \to D = \mathbb{P}^{1}$.
...

**66**

votes

**15**answers

23k 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.
...

**37**

votes

**8**answers

15k views

### Roadmap for studying arithmetic geometry

I have read Hartshorne's Algebraic Geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry.
I want to know how to use scheme ...

**80**

votes

**10**answers

8k 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 ...

**47**

votes

**3**answers

10k views

### Derived algebraic geometry: how to reach research level math?

I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different.
My goal is to study derived algebraic geometry, where derived ...

**120**

votes

**6**answers

14k 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 (...

**88**

votes

**7**answers

13k 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 one-...

**58**

votes

**1**answer

4k 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, http://www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined ...

**37**

votes

**6**answers

8k views

### Why are local systems and representations of the fundamental group equivalent

My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally ...

**53**

votes

**5**answers

14k 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 ...

**54**

votes

**12**answers

5k 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 ...

**40**

votes

**2**answers

3k 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)
...

**33**

votes

**8**answers

6k 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 $\mathcal{G}...

**39**

votes

**2**answers

4k views

### Ring-theoretic characterization of open affines?

Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...

**29**

votes

**6**answers

6k views

### Algebraic stacks from scratch [closed]

I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...

**64**

votes

**4**answers

5k views

### When is a singular point of a variety ($\mathcal{C}^\infty$-) smooth?

If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals.
The converse is false for a silly reason : in the real or ...

**14**

votes

**2**answers

5k views

### The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...

**38**

votes

**3**answers

2k views

### The Origin(s) of Modular and Moduli

In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/...

**39**

votes

**1**answer

16k views

### What is inter-universal geometry?

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...

**27**

votes

**4**answers

4k views

### The Jouanolou trick

In Une suite exacte de Mayer-Vietoris en K-théorie algébrique (1972) Jouanolou proves that for any quasi-projective variety $X$ there is an affine variety $Y$ which maps surjectively to $X$ with ...

**21**

votes

**4**answers

4k views

### When is an irreducible scheme quasi-compact?

The standard examples of schemes that are not quasi-compact are either non-noetherian or have an infinite number of irreducible components. It is also easy to find non-separated irreducible examples. ...

**21**

votes

**1**answer

2k views

### What is the precise relationship between Langlands and Tannakian formalism?

As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept!
In any case, I wish ...

**36**

votes

**1**answer

3k views

### A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives

The question below is the follow-up of this question on MathOverflow.
Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are ...

**23**

votes

**6**answers

2k views

### Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'.
Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...

**22**

votes

**4**answers

4k views

### Extending vector bundles on a given open subscheme

Let $U$ be a dense open subscheme of an integral noetherian scheme $X$ and let $E$ be a vector bundle on $U$. Suppose that the complement $Y$ of $U$ has codimension $\textrm{codim}(Y,X) \geq 2$. Let $...

**31**

votes

**8**answers

3k views

### Why do we need model categories?

I cannot give a good answer to this question. And
2) Why this definition of model category is the right way to give a philosophy of homotopy theory? Why didn't we use any other definition?
3) Has ...

**40**

votes

**2**answers

2k 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 ...

**18**

votes

**3**answers

2k views

### Are there “motivic” proofs of Weil conjectures in special cases?

This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil ...

**16**

votes

**3**answers

2k views

### Tangent space of Hilbert scheme

We have the following theorem:
Let $X$ be a projective scheme over an algebraically closed field $k$, and $Y \subset X$ a closed subscheme with Hilbert polynomial $P$. Then$$T_{[Y]}\text{Hilb}_P (X) =...

**32**

votes

**2**answers

820 views

### Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?

There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element".
Probably the best known analogy supporting that heuristic is the limit $q\to1$ for number ...

**27**

votes

**2**answers

2k 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 ...

**18**

votes

**3**answers

3k views

### What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...