Questions tagged [ag.algebraic-geometry]

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

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36 votes
1 answer
4k 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. ...
67 votes
6 answers
9k views

Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?
45 votes
2 answers
5k views

What interesting/nontrivial results in Algebraic geometry require the existence of universes?

Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that ...
37 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 ...
31 votes
6 answers
3k views

Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'. Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...
18 votes
3 answers
1k views

Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
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10 votes
1 answer
4k views

Bijection implies isomorphism for algebraic varieties

Let $f:X\to Y$ be a morphism of algebraic varieties over $\mathbb C$. Assume that a) $f$ is bijective on $\mathbb C$-points b) $X$ is connected c) $Y$ is normal. Does it imply that $f$ is an ...
237 votes
37 answers
152k views

Best algebraic geometry textbook? (other than Hartshorne)

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 ...
15 votes
2 answers
2k views

Explaining Mukai-Fourier transforms physically

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality). The basic algorithm is ...
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72 votes
5 answers
11k views

Is there a "geometric" intuition underlying the notion of normal varieties?

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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16 votes
2 answers
7k views

The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...
290 votes
8 answers
137k views

Philosophy behind Mochizuki's work on the ABC conjecture

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 ...
76 votes
9 answers
19k views

Results that are widely accepted but no proof has appeared

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....
47 votes
8 answers
7k views

When are there enough projective sheaves on a space X?

This question is being asked on behalf of a colleague of mine. Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...
37 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 $\...
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14 votes
1 answer
653 views

Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?

The q-Vandermonde identity reads: $$ \binom{m + n}{k}_{\!\!q} =\sum_{j} \binom{m}{k - j}_{\!\!q} \binom{n}{j}_{\!\!q} q^{j(m-k+j)} $$ The q-binomial coefficients: $$ \binom{ a }{ b}_{\!\!q} $$ ...
11 votes
2 answers
3k views

Conditions under which a bijective morphism of quasi-projective varieties is an isomorphism

I'm currently reading a paper by Nakajima (Quiver Varieties and Tensor Products), and I'm having a hard time understanding a very specific step in his proof of Lemma 3.2. Essentially, we have two (...
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7 votes
1 answer
1k views

When do two non-degenerate quadratic forms give rise to isomorphic Lie algebras?

Let $V$ be a vector space over some number field $k$. (I'm fine with $\mathbb{Q}$.) Let $\phi \colon V \to k$ be a non-degenerate quadratic form. Associated with $\phi$ is the orthogonal group $\...
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132 votes
6 answers
20k views

what mistakes did the Italian algebraic geometers actually make?

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 (...
96 votes
10 answers
12k views

equivalence of Grothendieck-style versus Cech-style sheaf cohomology

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 ...
96 votes
6 answers
10k views

Is there an analogue of curvature in algebraic geometry?

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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82 votes
15 answers
32k views

The importance of EGA and SGA for "students of today"

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. ...
73 votes
4 answers
6k views

When is a singular point of a variety ($\mathcal{C}^\infty$-) smooth?

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 ...
72 votes
16 answers
7k views

Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?

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 ...
51 votes
6 answers
7k views

Colimits of schemes

This is related to another question. I've found many remarks that the category of schemes is not cocomplete. The category of locally ringed spaces is cocomplete, and in some special cases this turns ...
50 votes
5 answers
13k views

The unification of Mathematics via Topos Theory

In her paper The unification of Mathematics via Topos Theory, Olivia Caramello says "one can generate a huge number of new results in any mathematical field without any creative effort". Is ...
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47 votes
8 answers
22k views

Roadmap for studying arithmetic geometry

I have read Hartshorne's Algebraic Geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. I want to know how to use scheme ...
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39 votes
8 answers
9k views

Ubiquity of the push-pull formula

The push-pull formula appears in several different incarnations. There are, at least, the following: 1) If $f \colon X \to Y$ is a continous map, then for sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}...
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
8 answers
5k 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 ...
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31 votes
4 answers
5k views

The Jouanolou trick

In Une suite exacte de Mayer-Vietoris en K-théorie algébrique (1972) Jouanolou proves that for any quasi-projective variety $X$ there is an affine variety $Y$ which maps surjectively to $X$ with ...
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28 votes
2 answers
8k views

When is fiber dimension upper semi-continuous?

Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$. When is this function ...
26 votes
1 answer
1k views

Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose: Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...
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24 votes
3 answers
4k views

When is an algebraic space a scheme?

Sometimes general theory is "good" at showing that a functor is representable by an algebraic spaces (e.g., Hilbert functors, Picard functors, coarse moduli spaces, etc). What sort of general ...
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21 votes
1 answer
2k views

naive de Rham cohomology fails for singular varieties

Let $X$ be a variety over a field $k$ of characteristic zero. If $X$ is smooth, algebraic de Rham cohomology defined as $$ H^n_{dR}(X / k)=\mathbb{H}^n(X, \Omega^\bullet_{X/k})\qquad (\star) $$ is a ...
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20 votes
4 answers
4k views

Extending vector bundles on a given open subscheme, reprise

In this question, Ariyan asks about the question of uniqueness of extensions of vector bundles when they exist. Sasha's answer suggests that extensions of vector bundles don't always exist. More ...
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20 votes
5 answers
6k views

A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite

I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb ...
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19 votes
4 answers
6k views

Isomorphism between varieties of char 0

Hi, the following statement appeared implicitly in a text I read and maybe you could just give me a hint how to see this resp. give a reference: If you have two k-varieties $X$ and $Y$ (sufficiently ...
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18 votes
2 answers
1k views

Explicit invariant of tensors nonvanishing on the diagonal

The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
  • 122k
16 votes
2 answers
2k views

Good introductory references on moduli (stacks), for arithmetic objects

I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and ...
  • 1,548
16 votes
3 answers
5k views

What are the current breakthroughs of Geometric Complexity Theory?

I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods. That program seems ...
14 votes
3 answers
2k views

Does isomorphic generic fibre imply isomorphic special fibre for smooth morphisms?

Let $X$ and $Y$ be regular integral Noetherian schemes. Assume that $X$ and $Y$ are smooth and proper over a base scheme $S=Spec R$, where $R$ is a discrete valuation ring. If $X$ and $Y$ have ...
13 votes
2 answers
2k views

How to prove that a projective variety is a finite CW complex?

Let $X$ be a (singular) projective variety, in other words something given by a collection of polynomial equations in $\mathbb CP^n$ or $\mathbb RP^n$. How can one prove it is a finite $CW$ complex? ...
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12 votes
1 answer
964 views

Schemes with no nonconstant maps to lower dimensional schemes

Fix an algebraically closed field $k$ (arbitrary characteristic), all schemes will be of finite type over $k$. (Property *): I'm interested in (classes of) examples of schemes $X$ (irreducible, of ...
  • 3,405
11 votes
2 answers
2k views

Global sections of flat scheme also flat?

In the most naive form my question would be as follows: If $f:X\to \mathrm{Spec}\;A$ is a flat morphism of schemes is it true that $H^0(X,\mathcal{O}_X)$ is a flat $A$-module? In general the answer ...
9 votes
1 answer
1k views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
9 votes
2 answers
1k views

Reference request on birational invariance of Chow group of zero cycles of degree zero

Let $CH_0(X)^0$ denote the group of zero cycles of degree zero modulo rational equivalence. I am looking for a reference for the following fact: If $X$ and $Y$ are smooth and projective varieties ...
  • 439
9 votes
1 answer
1k views

Obtaining non-normal varieties by pushout

In his answer to this MO question, Karl Schwede claimed that every non-normal variety can be obtained by an appropriate pushout diagram, as sketched in that answer. This would give substance to the ...
  • 21.9k
9 votes
0 answers
538 views

Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series; 1) multiplicative inversion for quadratic algebras, 2) compositional inversion for quadratic operads, 3) ...
8 votes
1 answer
769 views

When is the kernel of the etale fundamental group in a fibration abelian?

Let $X \to Y$ be a smooth proper morphism. Let $y$ be a geometric point of $Y$. Is the kernel of the natural map of etale fundamental groups $\pi_1^{et}(X_y) \to \pi_1^{et} (X)$ abelian? This is true ...
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