All Questions
22,546 questions
11
votes
0
answers
849
views
Infinite-dimensional affine space in algebraic geometry and algebraic topology
In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about ...
- 111
5
votes
1
answer
250
views
$\ell$-adic analogue of Kedlaya–Mochizuki
There is a well-known analogy between holonomic $\mathcal{D}$-modules on complex algebraic varieties and $\ell$-adic perverse sheaves on varieties over finite fields. Many theorems in one setting have ...
- 12.9k
0
votes
0
answers
55
views
Hom functor and Cohen-Macaulay modules
Let $A$ be a local Gorenstein $\mathbb{C}$-algebra (not necessarily regular). Let $M,N$ be maximal Cohen-Macaulay $A$-modules. Is Hom(M,N) a maximal Cohen-Macaulay A-module?
Note that I had asked this ...
- 1,040
6
votes
1
answer
142
views
Obstruction theory for specializing perfect complexes?
I'm considering a problem around the moduli of perfect complexes.
Let's consider a $X$ smooth proper over Spec($R$), $R$ is a DVR, $char=0$, equipped with a perfect complex $\mathcal F$ on $X_{K(R)}$.
...
- 16.2k
0
votes
0
answers
69
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 6,038
4
votes
0
answers
59
views
An algebraic group $G$ over $L^+$ such that $G_{\mathbb{R}}$ is compact for almost all embeddings
Does there exist a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is ...
- 12.9k
2
votes
0
answers
123
views
Action of torus on Laurent polynomials
Let $F$ be an algebraically closed field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for ...
- 376
1
vote
0
answers
61
views
Computing with the Picard group of non-integral curves
Let $C$ be a curve over $\mathbb{Q}_p$ and let $\mathcal{C}$ be a regular model of $C$ over $\mathbb{Z}_p$, with $\mathcal{J}$ the Neron model of the Jacobian of $C$. Raynaud's theorem asserts that $\...
- 111
3
votes
0
answers
53
views
Semisimple elements and fixed points
The following statement seems to be well-known:
Let $X$ be a variety on which an affine algebraic group $H$ acts with
finitely many orbits and let $s \in H$ be semisimple. Then $H_s = \{h
\in H \mid ...
- 53
5
votes
0
answers
126
views
+50
Dimension of the intersection of the commuting variety with a particular subspace
Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as:
$$
\mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}.
$$
It is well known that $\...
- 269
4
votes
1
answer
188
views
Projective automorphisms of a plane cubic curves
Let $E$ be a smooth cubic curve in $\mathbb{P}^2$ over an algebraically closed field $k$.
What is the group of the projective transformations preserving $E$ ?
In characteristic $0$ the answer is known ...
- 486
16
votes
5
answers
2k
views
"Classical" consequences of Bezout's theorem in dimensions $>2$
By Classical I mean something that could have been found before 1900 (say).
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
- 5,359
2
votes
0
answers
98
views
Connectedness of equivariant Hilbert schemes of points of affine spaces (or as orbifolds)?
Let $G$ be an abelian finite group act on $\mathbb C^n$, when the equivariant Hilbert scheme $\mathrm{Hilb}^{R}(\mathbb C^n)^G=\mathrm{Hilb}^{R}([\mathbb C^n/G])$ is connected? Now $R$ is a ...
- 249
14
votes
4
answers
2k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
- 28.2k
1
vote
1
answer
137
views
About dimensions of quotients of quasi projective varieties
This question is related to this one. If I have an locally closed, quasi projective scheme $X$ contained in an affine space, and a linearly reductive group $G$ acting freely on $X$, are there examples ...
- 3,720
3
votes
1
answer
1k
views
When may function (meromorphic) be expanded as power series with coefficients of integers?
Let $F$ be meromorphic function. With what properties may it be expanded as power series with coefficients of integers in such a form
$$
F=\sum_0^{\infty}a_i x^i,a_i\in \mathbb{N} \cup \{0\},\exists M ...
- 6,367
1
vote
0
answers
100
views
Examples of nontrivial morphism between simple bundles but not isomorphism
We know stable bundles have a good property:
If $f: E\longrightarrow E'$ is a nontrivial morphism where $E,E'$ have the same rank and degree, then $f$ must be an isomorphism.
I'm wondering does this ...
- 111
6
votes
1
answer
654
views
Is decomposability of polynomials over a field an undecidable problem?
By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as
$$
F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]),
$$
which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...
- 310
5
votes
1
answer
240
views
Galois action on Borovoi's algebraic fundamental group
In Borovoi's paper Abelian Galois cohomology of reductive groups, the algebraic fundamental group of a connected reductive group $G$ over a field $K$ of characteristic zero is defined as
$$\pi_1(G, T):...
- 14.2k
18
votes
1
answer
1k
views
Application of higher categories in algebra
Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher ...
CommunityBot
- 1
1
vote
0
answers
112
views
Zariski Connectedness Theorem: From Analytic & Topological Viewpoint
Let $p:Y \to X$ be a proper, flat (see later why) surjective map between smooth connected complex varieties $Y,X$, esp. $X$ unibranch. Assume that there exist a Zariski open (esp. dense) $U \subset X$ ...
- 6,038
3
votes
1
answer
143
views
Whitney stratifications of hypersurfaces
Suppose that $X$ is a Whitney stratified algebraic variety with strata $\{S_i\}.$
Suppose that $Z$ is a hypersurface of $X$ which transversely intersects all strata of $X$, i.e. $S_i \cap Z$ is a ...
- 5,544
3
votes
1
answer
469
views
Pushforward of functions on a frame bundle
Apologies in advance for the long setup and question.
Let $L \to X$ be a line bundle. We may take its frame bundle $p \colon Fr(L) \to X$, a $\mathbb{G}_m$-torsor. We have
$$ p_*\mathcal{O}_{Fr(L)} =...
CommunityBot
- 1
2
votes
0
answers
136
views
Effective Bombieri-Lang conjecture
The Bombieri-Lang conjecture is the following well-known conjecture:
Let $X$ be a projective variety defined over a number field $K$. Suppose that $X$ is general type. Then $X(K)$, the set of $K$-...
- 26.9k
3
votes
1
answer
225
views
Finite generativity of algebra with valuation
Let $C$ be a commutative finitely generated algebra with no zero divisors. If necessary, we can assume it to be graded and a unique factorization domain. Let $a\in C$ be a prime element.
Let's also ...
- 12.9k
29
votes
3
answers
4k
views
Have Grothendieck's notes in Montpellier already been investigated?
Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...
- 72
2
votes
0
answers
52
views
Can we bound the degree of a one dimensional smooth compact leaf of a holomorphic foliation in terms of its genus?
Let $X$ be a smooth projective variety over the complex numbers with a fixed ample line bundle $H$. Suppose that $\cal F$ is a foliation in curves over $X$ (which may be singular).
Can you find a ...
- 388
3
votes
1
answer
316
views
Which abelian varieties over a local field can be globalized?
As the title says, if $\mathcal{A}$ is an abelian variety over $\mathbb{Q}_p$, is there a criterion as to if I should expect there to exist $A$ over $\mathbb{Q}$ such that
$$\mathcal{A}\cong A\times_{\...
- 21.4k
1
vote
0
answers
76
views
Can one decompose quasi finite morphism as a composition of an open immersion and a finite morphism?
Can one decompose quasi-finite separated morphism of schemes as a composition of an open immersion and a finite morphism?
I am willing to assume that all the involved schemes are Noetherian.
- 2,649
8
votes
1
answer
860
views
What is the smallest and "best" 27 lines configuration? And what is its symmetry group?
I was this past year working with a bright high-schooler on algebraic geometry following Reid's book Undergraduate Algebraic Geometry, and we got all the way to proving that there is at least one line ...
- 21.4k
3
votes
1
answer
387
views
How to compute the transfer maps for G-theory of Noetherian schemes
Let $k$ be a field and $R$ be the ring $k[x,xy,xy^2,xy^3]$. Let $X$ be $\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of $X$ along the maximal ideal $I$ of $R$ generated by $x,xy,xy^2,xy^3$.I ...
CommunityBot
- 1
1
vote
0
answers
72
views
Hasse principle for Brauer groups of fields of transcendence degree 2
In his paper "A Hasse principle for function fields over PAC fields" (DOI link), Ido Efrat proves the following result: Let $F$ be an extension of a perfect PAC field $K$ of relative ...
- 63.9k
0
votes
0
answers
52
views
A question of irreducibility of certain affine algebraic sets
Let $K$ denote an algebraically closed field of characteristic zero, and let $p_1(T), \dots, p_m(T)$ denote $m$ irreducible polynomials in $K[x_1, \dots, x_n][T]$ of degree at least $1$. Set
$$ S= \{ (...
- 12.9k
0
votes
0
answers
78
views
Chow moving lemma with additional property
All varieties are over algebraically closed field of characteristic zero. Let $S$ be a smooth projective surface. Let $D$ be an irreducible divisor on $S$, $H$ be another divisor and $Z\subset S$ be a ...
- 219
1
vote
1
answer
881
views
Direct image of reflexive sheaf via finite, flat map
Suppose $f: X \rightarrow Y$ is a finite, flat (hence locally free) morphism of curves (i.e. schemes of dimension 1, not smooth or even reduced). Suppose $L$ is a reflexive sheaf on $X$, locally free ...
- 1,823
1
vote
0
answers
80
views
The definition of Hodge bundles with metric
A system of Hodge bundles is a direct sum of holomorphic vector bundles $E = \oplus_{p+q=n} E^{p,q}$ with a morphism $\theta : E^{p,q} \rightarrow E^{p-1,q+1} \otimes \Omega_X^1$ such that $\theta^2 = ...
- 11
0
votes
0
answers
49
views
The relation between Hodge bundles with metric and polarized variation of Hodge structures
Recently I've been reading Simpson's paper "constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, 1988, JAMS". On page 898 he mentioned about ...
- 11
3
votes
0
answers
135
views
Galois cohomology and Levi subgroups
Let $F$ a field and $G$ a smooth connected reductive group with a Levi subgroup $M$. Under what assumptions is $H^1(F, M) \to H^1(F, G)$ injective? In the case $F$ is nonarchimedean local I believe ...
- 605
0
votes
1
answer
204
views
Equivalence of dihedral and symmetric group actions on a specialized real algebra
Edit: fixed misaligned indentation for "Update x and y by", below. I also had two little ideas that might help.
consider first the case where the digit 7 is not allowed, simplifying the ...
CommunityBot
- 1
5
votes
1
answer
236
views
Explicit Jacquet-Langlands correspondence for real reductive groups
Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $...
- 761
11
votes
1
answer
411
views
Grothendieck purity for Brauer groups of stacks
Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
- 388
7
votes
1
answer
816
views
How to prove that topological Hochschild homology of a smooth proper stable k-linear infinity category is dualizable?
Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module ...
- 15.9k
2
votes
1
answer
135
views
Properness of quotient map
I am new to algebraic spaces and stacks. My question is the following:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Let $[X/G]$ be the quotient stack. Then when the natural map $\pi: ...
- 3,720
0
votes
0
answers
71
views
Reducible quartic space curve that is set-theoretic complete intersection
$\newcommand\P{\mathbb P} \newcommand\C{\mathbb C}$For a degree 4 curve, the Macaulay quartic curve is the only irreducible space curve (up to isomorphism) for which it remains an open problem to find ...
- 2,275
3
votes
1
answer
202
views
Square root of relative Kähler differentials and families of curves
Let $f: X \to S$ be a smooth morphism of schemes of relative dimension $1$. Then $K=\Omega^1_{X/S}$ is a line bundle on $X$. I am interested in the following question:
When does $\Omega_{X/S}$ have a ...
- 12.9k
-5
votes
0
answers
126
views
Is a quiver variety a moduli stack of quiver representations?
As the title, I was just wondering is a quiver variety a moduli stack of quiver representations? I am familiar with quiver varieties but not that familiar with moduli stacks, so I was just wondering ...
- 133
4
votes
0
answers
267
views
Is a pro-algebraic group over $\mathbb{Q}_p$ with Galois action the inverse limit of Galois-equivariant quotients?
Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...
- 15.4k
7
votes
2
answers
2k
views
The discriminant for the plane cubic curve
10 coefficients determine a degree 3 homogenous polynomial in $k[x,y,z]$. I understand that there is a degree 12 polynomial in these coefficients, called the discriminant, with 2040 terms, which ...
- 4,745
2
votes
0
answers
122
views
Polynomial discriminant equation
This is a fairly straightforward question, and I am hoping a definitive answer exists.
Does there exist a quadratic form $A \in \mathbb{C}[x_1, x_2, x_3, x_4]$ and a cubic form $B \in \mathbb{C}[x_1, ...
- 26.9k
394
votes
115
answers
110k
views
Not especially famous, long-open problems which anyone can understand
Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list in ...
- 24.2k