# Questions tagged [ag.algebraic-geometry]

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

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

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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 ...

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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 ...

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I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...

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Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...

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I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons (...

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This is not a very precise question, but I hope it will get some good answers.
As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign ...

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Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...

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Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is ...

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Sorry for that this is not a real question. But I thought people would like to know.
Alexandre Grothendieck died today: http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-...

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(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmuller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $G_{\...

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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 (...

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I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

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Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be ...

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The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm ...

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I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The ...

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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-...

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Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over ...

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This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference.
Here are two examples of the kind of combinatorial abstractions of geometric ...

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Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...

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I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...

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Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...

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This is a soft question.
I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...

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Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...

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This question was previously asked on Math SE.
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \...

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I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...

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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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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.
...

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The unsolvability of a general quintic equation in terms of the basic arithmetic operations and $n$th roots (i.e. the Abel–Ruffini theorem) is considered a major result in the mathematical canon. I ...

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I tried to answer an earlier question as to uses of GRR, just from my reading, although i do not understand GRR. Today i tried to understand the possible idea behind GRR. After editing my answer ...

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Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...

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I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...

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What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)?
I'd prefer examples that ...

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The background of this question is the talk given by Kevin Buzzard.
I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here....

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A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic ...

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In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are....

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Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...

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Let me start with three examples to illustrate my question (probably vague; I apologize in advance).
$\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M,...

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I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...

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There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-...

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If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals.
The converse is false for a silly reason : in the real or ...

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I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...

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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 ...

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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 ...

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Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories.
Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious ...

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I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...

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Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...

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This question is partly motivated by Never appeared forthcoming papers.
Motivation
Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance ...

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Call a category $C$ rigid if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example sets,...

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Griffiths' and Harris' book Principles of Algebraic Geometry is a great book with, IMHO, many typos and mistakes. Why don't we collaborate to write a full list of all of its typos, mistakes etc? My ...

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One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in ...