# Tagged Questions

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109 views

### Why “Fourier”-Mukai? [duplicate]

The Fourier-Mukai functor is one of the most important tools to work with in the derived category. While it is clear why the name of S.Mukai appears there,
why does Joseph Fourier appear in the name ...

**4**

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**1**answer

373 views

### Basics on anabelian geometry and Grothendieck's section conjecture

Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask ...

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560 views

### L-functions and algebraic geometry

Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection ...

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**2**answers

384 views

### Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...

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**1**answer

316 views

### What does it mean for a Deligne-Mumford stack to have trivial generic stabilizers?

I have stumbled upon some literature on Deligne-Mumford stacks, and it seems to me, at least superficially, that there is a strong link between DM-stacks which have "trivial generic stabilizers" and ...

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**1**answer

281 views

### Examples of Quot schemes

I'm studying Quot schemes, that I denote with $Quot_{N,X,P}$, with $N \in \mathbb{Z}$, $X \subset \mathbb{P}^d$ and $P \in \mathbb{Q}[t]$. So, I'm looking for explicit examples of Quot schemes. Could ...

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428 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 ...

**20**

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**4**answers

1k views

### Why Cohen-Macaulay rings have become important in commutative algebra?

I want to know the historic reasons behind singling out Cohen-Macaulay rings as interesting algebraic objects.
I'm reviewing my previous lecture notes about Cohen-Macaulay rings because now I'm ...

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453 views

### What is the meaning of “algebraic construction”, and how could this be used in algebraic geometry

I try to make my question clear:
When reading a paper or listening a seminar talk, people showed me some set, and claim it to be a scheme; or some map, and claim it to be a morphism. I query why this ...

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740 views

### Strict applications of deformation theory in which to dip one's toe

I hesitate to ask a question like this, but I really have tried finding answers to this question on my own and seemed to come up short. I readily admit this is due to my ignorance of algebraic ...

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1k views

### Is there a scheme corresponding to the unit interval?

Can someone complete the following table?
$\begin{array}{cc} \text{Topology over } \mathbb{R} & \text{Topology over } \mathbb{C} & \text{Algebraic Geometry} \\\\ \hline \mathbb{R} & ...

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1k views

### motivating geometric representation theory

I am wondering if there is a good motivation for geometric representation theory from within the questions of classical representation theory.
In other words, I'd be curious to see something using ...

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**2**answers

211 views

### Equivalent definitions of ample bundles

M. Atiyah in "VECTOR BUNDLES OVER AN ELLIPTIC CURVE" defined ample line bundle $E$ on $X$ as satisfying the following conditions:
Canonical map $H^0(X, E)\to E_x$ is surjective for any $x\in X$.
...

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**8**answers

1k views

### Self-containing structures

(This question is partly inspired by What is inter-universal geometry?.)
I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description ...

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798 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 ...

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**1**answer

950 views

### How many proofs of the Weil conjectures are there?

I hope this this is not seen as too much as jumping on the band-wagon, but here goes.
Deligne's proof of the last of the Weil conjectures is well-known and just part of a huge body of work that has ...

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**3**answers

1k views

### Intuitive pictures in characteristic p

This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...

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**1**answer

1k views

### How should a number theorist learn a modest amount of algebraic geometry?

A little bit vague, but I hope useful for the entire community.
I am, by training, an analytic number theorist. I have managed to learn some algebraic geometry, by reading parts of Silverman's ...

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**2**answers

1k views

### What conjectures in anabelian geometry are false?

Proving things suspected to be true in anabelian geometry is usually very hard. Maybe it is easier to disprove things suspected to be false?
In particular, I am interested in false generalizations of ...

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**0**answers

268 views

### Stability conditions for coherent sheaves and GIT

I am learning stability conditions for derived categories of coherent sheaves, following Bridgeland, and coming from a vector bundles background. $\mu$-stability for vector bundles has a clear GIT ...

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2k views

### Who will write the algebraic geometry texts that are needed? [closed]

Readers of MO are probably aware of the pedagogic need that would provoke such a query. It's around 60 years since Serre's FAC, and I imagine some people would say "you still have to read the original ...

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222 views

### Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?

The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it ...

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**2**answers

1k views

### Schemes and meaning of “geometric intuition”

I recently started studying algebraic geometry together with a couple of friends and especially in discussions online we keep reading about developing geometric intuition. There are some questions on ...

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5k views

### How has modern algebraic geometry affected other areas of math?

I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, ...

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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, ...

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**1**answer

315 views

### Higher direct images and singularities

Hi,
this is more or less a "reference" question.
Suppose $D$ a redueced irreducibile divisor in $X$ and I take $f:Y\rightarrow D$ a desingularization of his.
What information can i get from the ...

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votes

**1**answer

394 views

### Minimal resolution of Log del Pezzo surfaces

Suppose $X$ is a log del pezzo projective surface of index $l$. As far as I understand it will have a finite number of singular points all of which can be resolved by sucessive blow-ups.
Let $E_i$ be ...

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2k views

### Why are noetherian rings such natural objects in algebraic geometry?

I assume it is partially because they are good generalizations of polynomial rings, but what makes this generalization better than graded algebras or other generalizations of polynomial rings?

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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 ...

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**1**answer

439 views

### Characterization of algebraic points on Shimura varieties?

Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points
on Shimura varieties?
The question of course does not always make sense
for ${\bf{Q}}$-points: a theorem of Shimura shows ...

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**2**answers

909 views

### Grothendieck Riemann Roch involving Higher K ?

As we know, Grothendieck Riemann Roch only involves $K_{0}$. Is there any work generalizing this formula to (Quillen's Higher K)? If there is, what is the meaning for such kind of formula?
Thanks in ...

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3k views

### Geometric meaning of Cohen-Macaulay schemes

What is the geometric meaning of Cohen-Macaulay schemes?
Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various ...

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411 views

### Geometric meaning of small extensions ?

Let $(A,\mathfrak{m}_A)$ be a local Artinian $k$-algebra with residue field $k$. Then the scheme $\mathrm{Spec}(A)$ can be loosely seen as a "fat point", or an "infinitesimal neighbourhood" of a point ...

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1k views

### Why are they called Spherical Varieties?

My understanding is if you have a homogeneous space $X = G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then you call $X$ spherical.
Someone ...

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**1**answer

404 views

### Simplified treatment of resolutions of complex analytic varieties?

According to the article of Hauser:
The Hironaka theorem on resolution of singularities http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/home.html
The existence of resolution of ...

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654 views

### triangulated/derived categories in Physics and algebraic geometry

Why do physicists care about the triangulated/derived categories?
I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror ...

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103 views

### Removing a hypersurface when applying the Representation theorem to prove Positivstellensatz with uniform denominators

Let $f$ and $g$ be positive definite forms in the polynomial ring ${\mathbb{R}}[x_0,\ldots, x_n]$ such that $\deg(g)$ divides $\deg(f)$. A generalization of a theorem by Reznick is that $g^N f$ is a ...

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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 ...

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**0**answers

2k views

### Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...

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1k views

### Where can I learn about Formal Schemes?

I am trying to learn formal schemes. I tried to read the section in Hartshorne but I don't get very far from there since things are not done quite explicitly enough, at least in my opinion. I cannot ...

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**5**answers

1k views

### When do we study maps into an object or from the object to another object?

In many Mathematical theories, to study an object, we usually consider the set of all maps from that object to some other object. For example, in differential geometry, we study the smooth maps from a ...

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### Serre's FAC versus Hartshorne as an introduction to sheaves in algebraic geometry

I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started ...

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2k views

### Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties

This really is two questions, but they are kind of related so I would like to ask them at the same time.
Question 1:
In a question asked by Amitesh Datta, BCnrd commented that it is important to ...

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**1**answer

3k views

### Grothendieck's mathematical diagram.

I was going through this article which appeared in the Notices of the AMS, and in it, there's a picture which shows a mathematical diagram drawn by Grothendieck. I would be delighted if anyone could ...

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2k views

### Separable and algebraic closures?

I have no intuition for field theory, so here goes. I know what the algebraic and separable closures of a field are, but I have no feeling of how different (or same!) they could be.
So, what are the ...

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**3**answers

2k views

### Motives versus Motifs

I was in Paris recently for a meeting about motives or motifs, and since I'm too jet lagged
for real work let me ask the following somewhat frivolous question. The word "motif" is
usually translated ...

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**1**answer

548 views

### Geometry for Anderson's motives?

Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category ...

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**8**answers

3k views

### What's the difference between a real manifold and a smooth variety?

I am teaching a course on Riemann Surfaces next term, and would like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular ...

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2k views

### Why are normal crossing divisors nice?

This question is going to be extremely vague.
It seems that wherever I go (especially about Grothendieck's circle of ideas) the higher-dimensional analogue of a curve minus a finite number of points ...

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3k views

### Geometric intuition for limits

I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects ...