0
votes
0answers
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
votes
1answer
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 ...
2
votes
2answers
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 ...
5
votes
2answers
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 ...
5
votes
1answer
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 ...
2
votes
1answer
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 ...
16
votes
0answers
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
votes
4answers
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 ...
1
vote
4answers
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 ...
19
votes
2answers
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 ...
16
votes
3answers
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} & ...
13
votes
4answers
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 ...
3
votes
2answers
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 $x\in X$. ...
20
votes
8answers
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 ...
7
votes
4answers
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 ...
9
votes
1answer
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 ...
17
votes
3answers
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 ...
22
votes
1answer
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 ...
8
votes
2answers
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 ...
4
votes
0answers
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 ...
2
votes
0answers
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 ...
6
votes
0answers
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 ...
8
votes
2answers
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 ...
38
votes
12answers
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, ...
28
votes
4answers
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, ...
3
votes
1answer
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 ...
2
votes
1answer
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 ...
14
votes
5answers
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?
16
votes
5answers
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 ...
4
votes
1answer
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 ...
15
votes
2answers
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 ...
28
votes
4answers
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 ...
9
votes
2answers
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 ...
9
votes
2answers
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 ...
7
votes
1answer
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 ...
7
votes
0answers
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 ...
0
votes
2answers
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 ...
54
votes
2answers
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 "R├ęcoltes et Semailles" has been cited on various occasions on this forum. See for instance the answers to this question or ...
20
votes
0answers
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 ...
14
votes
2answers
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 ...
4
votes
5answers
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 ...
6
votes
3answers
3k views

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 ...
17
votes
5answers
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 ...
18
votes
1answer
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 ...
9
votes
2answers
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 ...
11
votes
3answers
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 ...
12
votes
1answer
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 ...
22
votes
8answers
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 ...
13
votes
2answers
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 ...
23
votes
8answers
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 ...