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Questions tagged [ag.algebraic-geometry]

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

40
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2answers
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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 ...
174
votes
35answers
110k 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
2answers
6k 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 ...
267
votes
7answers
126k 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 ...
52
votes
6answers
6k views

Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?
32
votes
1answer
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. ...
53
votes
5answers
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
1answer
422 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: $$ \binom{ a }{ b}_{\!\!q} $$ ...
13
votes
2answers
756 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 $$\...
41
votes
8answers
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 ...
21
votes
2answers
5k views

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 ...
36
votes
1answer
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 $\...
8
votes
1answer
681 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 ...
2
votes
1answer
284 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 $\widetilde{X} \to D = \mathbb{P}^{1}$. ...
64
votes
15answers
22k 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. ...
36
votes
8answers
14k 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 ...
116
votes
6answers
13k 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
7answers
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-...
56
votes
1answer
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 ...
53
votes
5answers
13k 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 ...
38
votes
6answers
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
12answers
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 ...
39
votes
2answers
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
8answers
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}...
29
votes
6answers
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 ...
39
votes
1answer
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 ...
38
votes
3answers
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/...
21
votes
1answer
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 ...
23
votes
6answers
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 ...
16
votes
4answers
4k 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 ...
31
votes
8answers
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 ...
21
votes
4answers
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 $...
18
votes
3answers
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
3answers
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 ...
39
votes
2answers
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 ...
32
votes
2answers
790 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 ...
26
votes
2answers
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 ...
17
votes
4answers
4k views

A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite

I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb ...
17
votes
3answers
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 ...
11
votes
3answers
3k 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 ...
22
votes
1answer
540 views

q-Catalan numbers from Grassmannians

In this question by $q$-Catalan numbers I mean the $q$-analog given by the formula $\frac{1}{[n+1]_q}\left[{2n\atop n}\right]_q$. The polynomial $\left[{2n\atop n}\right]_q$ represents the class of ...
11
votes
3answers
1k 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 ...
10
votes
2answers
534 views

Good introductory references on moduli (stacks), for arithmetic objects

I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and ...
8
votes
1answer
680 views

Obtaining non-normal varieties by pushout

In his answer to this MO question, Karl Schwede claimed that every non-normal variety can be obtained by an appropriate pushout diagram, as sketched in that answer. This would give substance to the ...
24
votes
1answer
840 views

Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose: Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...
15
votes
3answers
993 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 ...
11
votes
3answers
1k views

algebraic de Rham cohomology of singular varieties

Hi, Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular cohomology $H^*_{...
18
votes
1answer
1k views

Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$

Numerical evidence suggests a conjecture that the number of points of certain elliptic curve over $\mathbb{F}_p$ is either $p$ or $p+2$ for $p$ of certain form. Let $p$ be prime of the form $p=27a^2+...
11
votes
1answer
777 views

Descent theorems for fundamental groups and groupoids?

Grothendieck in his 1984 "Esquisse d'un programme" (Section 2) wrote (English translation): " ..,people still obstinately persist, when calculating with fundamental groups, in fixing a single base ...
8
votes
2answers
715 views

Reference request on birational invariance of Chow group of zero cycles of degree zero

Let $CH_0(X)^0$ denote the group of zero cycles of degree zero modulo rational equivalence. I am looking for a reference for the following fact: If $X$ and $Y$ are smooth and projective varieties ...