All Questions
22,770 questions
40
votes
1
answer
10k
views
What actually is the idea behind the condensed mathematics?
Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that analogs in the individual fields are instead ...
- 1,525
40
votes
4
answers
3k
views
Why are Green functions involved in intersection theory?
I've been learning Arakelov geometry on surfaces for a while. Formally I've understood how things work, but I'm still missing a big picture.
Summary:
Let $X$ be an arithmetic surface over $\...
- 1,237
40
votes
1
answer
14k
views
Why is Faltings' "almost purity theorem" a purity theorem?
My understanding of purity theorems is that they come in several flavors:
1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob_p$ are algebraic numbers all of whose ...
- 585
40
votes
1
answer
5k
views
Clausen–Scholze's Theorem 9.1 of Analytic.pdf, in view of light condensed sets, AKA is the Liquid Tensor Experiment easier now?
In the recent lecture series run jointly from IHÉS and Bonn, Clausen and Scholze have reworked—again—their foundations of geometry to focus attention on not arbitrary condensed sets and solid modules ...
- 35.4k
40
votes
3
answers
3k
views
On what kind of objects do the Galois groups act?
I am neither number theorist nor algebraic geometer. I am wondering
whether Galois groups of number fields (say the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which
...
- 21.8k
40
votes
5
answers
4k
views
How should one approach tropical mathematics?
Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across Litvinov - The Maslov dequantization, idempotent and tropical ...
40
votes
1
answer
3k
views
Is every connected scheme path connected?
Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following.
Let's ...
- 47.5k
40
votes
1
answer
4k
views
A closed subscheme of an open subscheme that is not an open subscheme of a closed subscheme?
A morphism $f: V \rightarrow X$ of schemes is a locally closed immersion if it can be factored into a closed immersion followed by an open immersion. It is not hard to show that if $f$ is an open ...
- 3,857
40
votes
1
answer
2k
views
Implications and consequences of the recent proof of the geometric Langlands conjecture
I am a beginner in mathematical physics and geometric Langlands, having very limited knowledge in both fields so far.
The proof of geometric Langlands conjecture is published a few months ago. What ...
- 203
40
votes
1
answer
4k
views
Roadmap to Geometric Representation Theory (leading to Langlands)?
I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own.
I'm becoming increasingly fascinated by stuff ...
- 7,789
39
votes
7
answers
11k
views
Geometric meaning of the Euler sequence on $\mathbb{P}^n$ (Example 8.20.1 in Ch II of Hartshorne)
Is there any geometric way to understand the exact sequence in Example 8.20.1 in Ch II of Hartshorne (p. 182), or its dual from theorem 8.13?
Here is the sequence:
$0\to O_{\mathbb{P}^n}\to O_{\...
- 1,219
39
votes
6
answers
9k
views
What is the inverse image sheaf necessary for in algebraic geometry?
Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf
$$U \mapsto \...
- 7,328
39
votes
9
answers
5k
views
What is a deformation of a category?
I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.
What is a deformation of a (linear, dg, ...
- 21k
39
votes
6
answers
5k
views
Elementary examples of the Weil conjectures
I'm looking for examples of the Weil conjectures---specifically rationality of the zeta function---that can be appreciated with minimal background in algebraic geometry. Are there varieties for which ...
- 7,994
39
votes
1
answer
2k
views
$V$, $W$ are varieties. Does $V\times \mathbf{P}^1=W\times \mathbf{P}^1$ imply $V=W$?
If $\mathbf{P}^1$ is replaced by the affine line $\mathbf{A}^1$, this becomes the cancellation problem, and we have a pair of famous Danielewski surfaces ($xy=1-z^2$ and $x^2y=1-z^2$) as a ...
- 1,079
39
votes
8
answers
7k
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 ...
- 7,328
39
votes
6
answers
6k
views
Using algebraic geometry to understand class field theory
In Algebraic Number Theory, S. Lang says "[a geometrical approach] allows one to have a much clearer insight into the whole class field theory, since the existence theorem and
the reciprocity law ...
- 711
39
votes
5
answers
5k
views
What is the general geometric interpretation of modules in algebraic geometry?
Algebraic geometry is quite new for me, so this question may be too naive. therefore, I will also be happy to get answers explaining why this is a bad question.
I understand that the basic philosophy ...
- 2,027
39
votes
3
answers
10k
views
What does the term "yoga" mean in mathematics?
Just exactly what the title says; often, in mathematics, particularly in the vicinity of Grothendieck, I see reference to "the yoga of...". What exactly does the term "yoga" mean in these contexts?
- 5,827
39
votes
4
answers
10k
views
difference between equivalence relations on algebraic cycles
For the definitions of the equivalence relations on algebraic cycles see http://en.wikipedia.org/wiki/Adequate_equivalence_relation.
I want to know how far away from each other the equivalence ...
user19475
39
votes
6
answers
8k
views
Down-to-earth expositions of Hodge theory
What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory?
Of course, since I haven't found a (for me) readable introduction, I don't know what I ...
- 391
39
votes
5
answers
6k
views
Algebraic machinery for algebraic geometry
Hello everybody,
I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative ...
39
votes
4
answers
4k
views
What can we learn from the tropicalization of an algebraic variety?
I often hear people speaking of the many connections between algebraic varieties and tropical geometry and how geometric information about a variety can be read off from the associated tropical ...
- 11.6k
39
votes
3
answers
6k
views
What do higher Chow groups mean?
Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i,...
- 12.3k
39
votes
1
answer
5k
views
Flatness in Algebraic Geometry vs. Fibration in Topology
I am currently trying to get my head around flatness in algebraic geometry. In particular, I'm trying to relate the notion of flatness in algebraic geometry to the notion of fibration in algebraic ...
- 21.2k
39
votes
2
answers
5k
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. ...
- 1,589
39
votes
2
answers
6k
views
What is Serre's condition (S_n) for sheaves?
The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
- 30.5k
39
votes
4
answers
3k
views
Does a scheme have a "separification"?
Background:
(1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful ...
39
votes
5
answers
3k
views
Surfaces in $\mathbb{P}^3$ with isolated singularities
It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, ...
- 66.3k
38
votes
18
answers
24k
views
Learning about Lie groups
Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
38
votes
7
answers
7k
views
What is DAG and what has it to do with the ideas of Voevodsky?
In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...
- 1,085
38
votes
8
answers
6k
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 ...
- 1,040
38
votes
6
answers
6k
views
"Points" in algebraic geometry: Why shift from m-Spec to Spec?
Why were algebraic geometers in the 19th Century thinking of
m-Spec as the set of points of an affine variety associated to the
ring whereas, sometime in the middle of the 20 Century, people started ...
- 2,967
38
votes
6
answers
4k
views
Are all polynomial inequalities deducible from the trivial inequality?
I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of ...
- 118k
38
votes
5
answers
7k
views
Why does so much recent work involve K3 surfaces?
I've been noticing that a whole lot of papers published to the Arxiv recently involve K3 surfaces. Can anyone give me (someone who, at this point, knows little more about K3 surfaces than their ...
38
votes
7
answers
5k
views
Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)
In this question, Harry Gindi states:
The fact that a commutative ring has a natural topological space associated with it is a really interesting coincidence.
Moreover, in the answers, Pete L. ...
- 21k
38
votes
3
answers
8k
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 development ...
- 356
38
votes
2
answers
8k
views
Intuition for Primitive Cohomology
In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\wedge \eta$. We then ...
- 16k
38
votes
2
answers
4k
views
A geometric characterization for arithmetic genus
Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others):
the ...
- 7,328
38
votes
2
answers
3k
views
Do Grothendieck universes matter for an algebraic geometer?
I recently learned that some parts of SGA require axioms beyond ZFC. I am just a simple algebraic geometer so I am trying to understand how can this fact impact my life (you may have engaged in a ...
user138661
38
votes
2
answers
2k
views
Residues in several complex variables
I am trying to educate myself about the basics of the theory of residues in several complex variables. As is usually written in the introduction in the textbooks on the topic, the situation is much ...
- 1,190
38
votes
1
answer
2k
views
Why is there no Brauer scheme?
Let $X$ be a proper scheme over a base field $k$ (one could consider more general settings, but I am primarly interested in a "geometric" situation with $k$ being algebraically closed).
Then the ...
- 6,920
38
votes
1
answer
2k
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 $\...
- 2,027
38
votes
2
answers
3k
views
What about stacks of categories in algebraic geometry?
Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The ...
- 35.4k
38
votes
1
answer
2k
views
When do 27 lines lie on a cubic surface?
Consider $27$ (pairwise distinct!) lines in $\mathbb{P}^3$ whose intersection graph is that expected¹ of the $27$ lines on a smooth cubic surface. Question: Is there a simple necessary and sufficient ...
- 32.5k
38
votes
0
answers
1k
views
Computer calculations in A_infinity categories?
Is there a good computer program for doing calculations in A-infinity categories?
Explicit calculations in A-infinity categories are an important, useful, yet very tedious task. One has to keep track ...
- 1,439
37
votes
6
answers
6k
views
Why study Higher Sheaf Cohomology?
The classical lore is that $H^1(X,\mathcal F)$ is obstruction to lifting local data to global data. However I don't understand why one would want to compute $H^3(X,\mathcal F), H^4(X,\mathcal F), \...
- 517
37
votes
7
answers
8k
views
Model theoretic applications to algebra and number theory(Iwasawa Theory)
One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall ...
- 2,396
37
votes
7
answers
4k
views
Is an algebraic space group always a scheme?
Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme?
My feeling is that the answer is "yes" ...
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k