Questions tagged [ag.algebraic-geometry]

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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Are we better in computing integrals than mathematicians of 19th century?

When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
39 votes
7 answers
10k 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_{\...
Enrique Acosta's user avatar
39 votes
6 answers
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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 \...
Charles Staats's user avatar
39 votes
6 answers
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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 ...
Jonathan Wise's user avatar
39 votes
9 answers
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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, ...
Kevin H. Lin's user avatar
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$V$, $W$ 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 ...
Honglu's user avatar
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1 answer
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Clausen's modified Hodge Conjecture

In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online. If I'...
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5 answers
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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
6 answers
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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 ...
user47573's user avatar
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4 answers
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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 ...
J.C. Ottem's user avatar
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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 ...
user34143's user avatar
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4 answers
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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 ...
Anton Geraschenko's user avatar
38 votes
18 answers
23k 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.
Daniel Erman's user avatar
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38 votes
7 answers
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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 ...
veit79's user avatar
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8 answers
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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 ...
Megan's user avatar
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6 answers
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"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 ...
Randomblue's user avatar
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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 ...
Qiaochu Yuan's user avatar
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. ...
Kevin H. Lin's user avatar
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38 votes
8 answers
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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 ...
Charles Staats's user avatar
38 votes
3 answers
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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 ...
Ali Taghavi's user avatar
38 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?
Sridhar Ramesh's user avatar
38 votes
10 answers
6k views

Algebraic Geometry versus Complex Geometry

This question is motivated by this one. I would like to hear about results concerning complex projective varieties which have a complex analytic proof but no known algebraic proof; or have an ...
38 votes
13 answers
5k views

Exposition of Grothendieck's mathematics

As Wikipedia says: In Grothendieck's retrospective Récoltes et Semailles, he identified twelve of his contributions which he believed qualified as "great ideas". In chronological order, ...
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 ...
Charles Staats's user avatar
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 ...
user avatar
38 votes
1 answer
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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 ...
Daniel Loughran's user avatar
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 ...
user25309's user avatar
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38 votes
1 answer
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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. ...
Nathaniel Bottman's user avatar
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 $\...
KotelKanim's user avatar
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38 votes
5 answers
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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, ...
Francesco Polizzi's user avatar
37 votes
8 answers
12k views

What is the proper initiation to the theory of motives for a new student of algebraic geometry?

A preliminary apology is in order: I realize that most of my contributions to this site are in the form of reference requests. I understand that this makes it seem as though I do nothing more than sit ...
lambdafunctor's user avatar
37 votes
6 answers
5k 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 ...
Gabriel's user avatar
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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" ...
Anton Geraschenko's user avatar
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 ...
S. Carnahan's user avatar
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37 votes
2 answers
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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 ...
Charles Siegel's user avatar
37 votes
2 answers
1k views

varieties with points in number fields

Let $V$ be a projective variety, defined over $\mathbb{Q}$. Suppose that for every number field $K \neq \mathbb{Q}$, there is a $K$-point of $V$. Does it follow that $V$ has a $\mathbb{Q}$-point? ...
Eric Larson's user avatar
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37 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 ...
Bananeen's user avatar
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37 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 ...
Hailong Dao's user avatar
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37 votes
3 answers
3k views

What does it mean geometrically that an element in a domain is irreducible?

Consider a domain $A$ and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory ...
Georges Elencwajg's user avatar
37 votes
1 answer
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 ...
David Roberts's user avatar
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37 votes
2 answers
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Schubert calculus, as lowbrow as possible

Starting in a week I'm going to be an instructor at a summer program for exceptionally mathematically talented high school students, and I'm going to be teaching a class on Schubert calculus. The ...
Nicolas Ford's user avatar
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37 votes
1 answer
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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 ...
Gro-Tsen's user avatar
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37 votes
1 answer
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Morava on Shafarevich conjecture

$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture. The Shafarevich Conjecture states: ...
Romeo's user avatar
  • 2,714
37 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 ...
Heinrich Hartmann's user avatar
37 votes
0 answers
5k views

In memoriam Torsten Ekedahl [closed]

This is not a question, just very, very, very sad news. Our community has lost one of its most active members. I am posting a letter I received recently. From Carel Faber and Sandra Di Rocco: ...
36 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), \...
finnlim's user avatar
  • 507
36 votes
9 answers
5k views

Computing fundamental groups and singular cohomology of projective varieties

Are there any general methods for computing fundamental group or singular cohomology (including the ring structure, hopefully) of a projective variety (over C of course), if given the equations ...
Kevin H. Lin's user avatar
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36 votes
6 answers
8k views

What is the Zariski topology good/bad for?

In a comment to this question the quotation "The Zariski Topology is the 'Wrong' topology for Algebraic Geometry" appears. Well, so some spontaneous questions arise: 1) What is Zariski topology ...
36 votes
5 answers
4k 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 ...
KotelKanim's user avatar
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