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

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127
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33answers
74k views

Best Algebraic Geometry text book? (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 ...
25
votes
2answers
5k 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. Added : According to their ...
212
votes
7answers
108k 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 ...
40
votes
5answers
4k 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 ...
31
votes
1answer
2k 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. ...
44
votes
15answers
14k 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. ...
29
votes
8answers
3k 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 sheaf ...
29
votes
2answers
2k 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 ...
93
votes
6answers
9k 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 ...
41
votes
9answers
4k 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 ...
10
votes
1answer
797 views

geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring. What is the geometric interpretation of: 1- Gorenstein rings 2- Complete intersections 3- Regular rings? and how can I realize differences by geometric ...
30
votes
2answers
2k views

Is every Noetherian Commutative Ring a quotient of a Noetherian Domain?

This was an interesting question posed to me by a friend who is very interested in commutative algebra. It also has some nice geometric motivation. The question is in two parts. The first, as stated ...
16
votes
3answers
2k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
13
votes
2answers
653 views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
11
votes
3answers
976 views

What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?

More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...
3
votes
1answer
356 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 ...
8
votes
1answer
702 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 ...
6
votes
1answer
712 views

Regular monomorphisms of schemes

In the category of schemes, the equalizer of two morphisms $f,g : X \to Y$ is always a locally closed immersion into $X$ (since this is just $X \times_{Y \times Y} Y$ and $\Delta : Y \to Y \times Y$ ...
5
votes
1answer
293 views

Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...
7
votes
2answers
521 views

Is $G/T$ a projective variety?

Let $G$ be a semisimple Lie group and $T$ be its maximal torus. Can we say that $G/T$ is a projective variety?. Is there any proof or counterexample for it?
7
votes
1answer
525 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 ...
67
votes
2answers
15k views

Why is the Hodge Conjecture so important?

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 ...
49
votes
20answers
10k views

Algebraic geometry examples

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 ...
44
votes
11answers
6k views

Interesting results in algebraic geometry accessible to 3rd year undergraduates

On another thread I asked how I could encourage my final year undergraduate colleagues to take an algebraic geometry or complex analysis courses during their graduate studies. Willie Wong proposed me ...
56
votes
6answers
5k 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 ...
20
votes
17answers
12k 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.
42
votes
8answers
5k views

Questions about analogy between Spec Z and 3-manifolds

I'm not sure if the questions make sense: Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
36
votes
4answers
5k 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 ...
47
votes
3answers
3k views

Why do Todd classes appear in Grothendieck-Riemann-Roch formula?

Suppose for some reason one would be expecting a formula of the kind $$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$ valid in $H^*(Y)$ where $f:X\to Y$ is a ...
43
votes
5answers
4k views

Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?
39
votes
5answers
7k views

Why tropical geometry?

Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being ...
34
votes
9answers
5k views

What are dessins d'enfants?

There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led ...
43
votes
1answer
2k views

What is the relationship between motivic cohomology and the theory of motives?

I will begin by giving a rough sketch of my understanding of motives. In many expositions about motives (for example, www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined to be ...
21
votes
5answers
6k views

Textbook for Etale Cohomology

What is the best textbook (or book) for studying Etale cohomology?
29
votes
8answers
4k 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 ...
30
votes
2answers
2k views

How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)

What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me) ...
20
votes
6answers
4k views

Heuristic behind the Fourier-Mukai transform

What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform? Moreover, could someone recommend a concise introduction to the subject?
20
votes
5answers
2k 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 ...
19
votes
2answers
3k 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 ...
45
votes
3answers
2k views

Is “semisimple” a dense condition among Lie algebras?

The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some ...
15
votes
5answers
5k views

Nonsingular/Normal Schemes

I always had trouble remembering this. Is it true that a curve over a non-algebraically-closed field is normal implies that it's non-singular? How about a 1 dimensional scheme? How about dimension 2? ...
56
votes
4answers
3k views

Rigidity of the category of schemes

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 ...
37
votes
1answer
13k views

What is inter-universal geometry?

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
18
votes
4answers
2k views

When are GIT quotients projective?

Some background on GIT Suppose G is a reductive group acting on a scheme X. We often want to understand the quotient X/G. For example, X might be some parameter space (like the space of possible ...
9
votes
4answers
1k 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 ...
8
votes
1answer
3k views

A nice explanation of what is a smooth (l-adic) sheaf?

I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.
26
votes
1answer
2k views

Special values of L-functions as periods

If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles. For example, when ...
21
votes
5answers
2k views

Verlinde's formula

"Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT. Depending on... • which chiral CFT one considers (does one restrict to WZW models, or not?) ...
22
votes
7answers
2k views

Invariant Polynomials under a Group Action (hidden GIT)

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$). Now the symmetric group $\mathfrak{S}_n$ ...
20
votes
2answers
2k views

Criteria for irreducibility of polynomial

If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible? Thank you very much, best