All Questions
2,036 questions
27
votes
3
answers
5k
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 ...
- 1,990
22
votes
6
answers
8k
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 ...
- 14.3k
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 ...
- 211
18
votes
5
answers
1k
views
Number of $3\times 3$ anticommuting matrices over finite fields $\mathbb{F}_p$ is (or is not?) polynomial in $p$?
There are rare algebraic varieties such that the number of points over finite fields $\mathbb{F}_p$ is given by a polynomial in $p$. One notable series of examples is the commuting variety: $[A,B]=0$ ...
- 24.9k
18
votes
4
answers
7k
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 ...
- 183
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 ...
- 1,364
14
votes
3
answers
3k
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 ...
- 21.4k
14
votes
2
answers
3k
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?
...
- 28.9k
14
votes
3
answers
2k
views
Adelic description of moduli of $G$-bundles on a curve
Let $X$ be a smooth, projective, geometrically connected curve over a field $k$ and $G$ an an affine algebraic group group over $k$ (we can put more hypotheses on $G$ if necessary). If $K$ denotes the ...
- 3,605
13
votes
3
answers
2k
views
Is the map on étale fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?
$\DeclareMathOperator\Spec{Spec}$Let $k \subset L$ be two algebraically closed fields of characteristic $0$. Let $U \subset \mathbb P^n_k$ be a smooth quasi-projective variety and let $U_L$ denote the ...
- 1,308
13
votes
2
answers
3k
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 ...
- 1,645
13
votes
1
answer
1k
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 ...
- 3,555
11
votes
1
answer
2k
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 ...
- 23.4k
10
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
2k
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 ...
- 469
8
votes
1
answer
886
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 ...
- 149k
7
votes
1
answer
1k
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 $\Delta : Y \to Y \times Y$ ...
- 63.1k
3
votes
1
answer
551
views
Bounding the number of critical points in a Lefschetz pencil
Let $k$ be an algebraically closed field. Let $X/k$ be a smooth projective variety. For a suitable embedding in $\mathbb{P}^{n}$ we can form a Lefschetz pencil $\widetilde{X} \to D = \mathbb{P}^{1}$.
...
- 5,504
2
votes
1
answer
185
views
Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q) $, where $P_i$ - do not depend on $N$?
Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating ...
- 24.9k
182
votes
33
answers
32k
views
What should be learned in a first serious schemes course?
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 (...
154
votes
7
answers
85k
views
Where to buy premium white chalk in the U.S., like they have at RIMS? [closed]
While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek.
At RIMS (in Kyoto) in 2005, they had the best white ...
150
votes
31
answers
70k
views
What are the most misleading alternate definitions in taught mathematics?
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
108
votes
7
answers
21k
views
What is the field with one element?
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-...
- 5,062
103
votes
3
answers
6k
views
Why do combinatorial abstractions of geometric objects behave so well?
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 ...
- 24.2k
101
votes
2
answers
11k
views
Riemann hypothesis via absolute geometry
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 ...
- 5,232
86
votes
44
answers
21k
views
Demystifying complex numbers
At the end of this month I start teaching complex analysis to
2nd year undergraduates, mostly from engineering but some from
science and maths. The main applications for them in future
studies are ...
84
votes
1
answer
5k
views
Is there a complex surface into which every Riemann surface embeds?
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 \...
- 19.3k
78
votes
6
answers
6k
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 sets,...
- 63.1k
65
votes
5
answers
18k
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 ...
65
votes
4
answers
22k
views
When is the product of two ideals equal to their intersection?
Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds ...
- 914
65
votes
17
answers
17k
views
Good introductory references on algebraic stacks?
Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...
- 1,538
61
votes
11
answers
21k
views
What are some open problems in algebraic geometry?
What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...
55
votes
1
answer
4k
views
Does every smooth, projective morphism to $\mathbb{C}P^1$ admit a section?
Possibly this has already been asked, but it came up again in this question of Daniel Litt. Does every smooth, projective morphism $f:Y\to \mathbb{C}P^1$ admit a section, i.e., a morphism $s:\mathbb{...
- 4,111
54
votes
5
answers
2k
views
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge ...
- 10.1k
52
votes
2
answers
7k
views
Ring-theoretic characterization of open affines?
Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
- 5,407
52
votes
2
answers
4k
views
a categorical Nakayama lemma?
There are the following Nakayama style lemmata:
(the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m_1, \ldots, m_n$ generate $M$ ...
user19475
51
votes
22
answers
19k
views
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
50
votes
2
answers
5k
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)
...
- 5,515
43
votes
1
answer
19k
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 ...
- 10.8k
42
votes
3
answers
11k
views
What does the Lefschetz principle (in algebraic geometry) mean exactly?
This principle claims that every true statement about a variety over the complex number field $\mathbb{C}$ is true for a variety over any algebraic closed field of characteristic 0.
But what is it ...
- 1,102
42
votes
3
answers
5k
views
The Origin(s) of Modular and Moduli
In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/...
- 10.5k
42
votes
2
answers
3k
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 ...
41
votes
3
answers
3k
views
Can the unsolvability of quintics be seen in the geometry of the icosahedron?
Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...
- 151k
41
votes
7
answers
9k
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?
- 2,132
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 ...
- 35.5k
34
votes
3
answers
3k
views
What is the theory of local rings and local ring homomorphisms?
It is well-known that the category of local rings and ring homomorphisms admits an axiomatisation in coherent logic. Explicitly, it is the coherent theory over the signature $0, 1, -, +, \times$ with ...
- 15.9k
33
votes
4
answers
4k
views
How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?
As the question title asks for, how do others visualize the Riemann-Roch theorem with complex analysis or geometric topology considerations? That is all Riemann would have had back in the day, and he ...
32
votes
7
answers
5k
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 $\mathfrak{S}_n$ ...
- 1,993
31
votes
2
answers
2k
views
Why is the motivic category defined over the site of smooth schemes only?
Fix a base scheme $S$. Stable and unstable motivic categories over $S$ are defined as certain categories of higher stacks on the Nisnevich site $Sm_S$ of smooth schemes over $S$. Why smooth?
As a ...
- 64k
31
votes
5
answers
3k
views
Is there a Whitney theorem type theorem for projective schemes?
We know that any smooth projective curve can be embedded (closed immersion) in $\mathbb{P}^3$. By definition a projective scheme over $k$ admits an embedding into some $\mathbb{P}^n$. Can we create an ...
- 433