All Questions
32,954
questions
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16
views
Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$
Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
- 151
3
votes
1
answer
113
views
Are limits of compact leaves compact?
Let $M$ be a compact smooth manifold, and $\mathcal{F}$ be a foliation on $M$. Assume that $L$ is a leaf of $\mathcal{F}$ for which there is $x\in L$ with the property that every neighborhood of $x$ ...
- 1,093
0
votes
0
answers
43
views
Reference request: explicit formula for Lie derivative of matrix Lie groups
Let $M =End(n,\mathbb{C})$ be the space of complex matrices with adjoint action by $ U(n)$, i.e. acted by $B\rightarrow gBg^{-1}$ for $B\in End(n,\mathbb{C}),g\in U(n)$. Let $X_{\xi}$ be the vector ...
- 123
0
votes
0
answers
25
views
Wasserstein space isomorphic to original space?
Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$?
Note that there is a canonical non-...
3
votes
1
answer
200
views
"Noetherianess" of $\mathrm{Mod}(\mathbb{F}_{p,\square})$
In classical commutative ring theory it is quite immediate to see, that a field is noetherian in the following sense:
For any finitely generated $k$-vectorspace $M$, any sub-object is finitely ...
2
votes
0
answers
29
views
Connection vs Exponential preserving maps
Connection Preserving Diffeomorphisms
The setting is a manifold $M$ equipped with a linear connection $\nabla$. Kobayashi & Nomizu [K&N §VI.1] define a connection preserving diffeomorphism (...
- 243
1
vote
0
answers
26
views
Boundary behavior for submanifolds with bounded second fundamental form
I am interested in a boundary version of this question About hypersurfaces in R^n+1 with bounded 2nd fundamental form.
The question is as follows. Let $\Sigma^k\subset \Bbb R^n$ be a submanifold with ...
- 141
2
votes
1
answer
72
views
Compute Christoffel symbols of sphere by embedding
In his answer V. Semeria, starts by taking
$$(y_1,\dots,y_{n+1})=\left(x_1,\dots,x_n,\sum_{i=1}^{n+1}x_i^2 -R^2\right)$$
Write $(\vec{e}_1,\dots,\vec{e}_{n+1})$ the canonical basis of $\mathbb{R}^{n+1}...
- 21
1
vote
0
answers
77
views
Monomorphism which is locally of finite presentation
$\DeclareMathOperator\Spec{Spec}$Let $X$ be an affine scheme of finite type over a field. Let $Y\to X$ be a monomorphism of schemes, which is locally of finite presentation. Is it true that $f$ is ...
- 11
2
votes
1
answer
77
views
Construction of Scherk's surface using soap films
I am currently interested in the differential geometry of minimal surfaces, and I have a rather trivial question regarding Scherk's surface (the one which can be parametrised by the real function $(x,...
- 506
2
votes
1
answer
124
views
Artin's "Autoduality of the Jacobian"
In some of his papers (for example, in "Formal groups arising from algebraic varieties" with B. Mazur), M. Artin cites
M. Artin and B. Wyman, Autoduality of the Jacobian, Bowdoin College, ...
- 995
2
votes
1
answer
161
views
Infinitesimal neighborhood and Ext group
$\DeclareMathOperator\Ext{Ext}$Let $X$ be a smooth projective complex variety and $\iota\colon Y\subset X$ be a smooth closed subvariety. It is well-known that there is a spectral sequence
$$E_2^{p,q}=...
- 131
3
votes
1
answer
208
views
Endomorphism ring of a generic elliptic curves in positive characteristic
Let E be a generic elliptic curve over an algebraically closed field $k$ of characteristic $p>0$ (i.e. an elliptic curve corresponding to a geometric point over the generic point of $M_{1,1}$).
...
- 177
1
vote
1
answer
126
views
Two different resolution of a three fold
Let $X\subset \mathbb{A}^4_\mathbb{C}$ be a three fold defined by the equation $xy-zw=0$
This variety has a singularity at origin of $\mathbb{A}^4_\mathbb{C}$
If we blow up this three fold in two ways ...
- 83
2
votes
1
answer
132
views
How to decompose a given polynomial by ideal generators
Given a finite set of polynomials $f_1, f_2,..., f_n$ of variables $x_1,...,x_m$, generating the ideal $I$, suppose that we have one more polynomial $g\in I$.
What is the algorythm for decomposing $g$ ...
- 621
2
votes
0
answers
37
views
Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonnegative curvature operator
Let $S^m$ be a standard sphere of dimension $m=n+4k$, and let $M$ be any closed Riemaninan manifold of dimension $n$ with nonnegative curvature operator.
My question: Is there always a smooth spin map ...
- 553
1
vote
1
answer
66
views
For proper group action on closed Riemannian manifold, must the union of orbits with non-unique closest points to a given point be of 0 volume measure
Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}d\operatorname{vol}_g \forall E \in \mathcal{B}(M),$ the ...
- 1,417
1
vote
0
answers
66
views
Question from Taubes' SW$\Rightarrow$ Gr
I am trying to understand Taubes' paper on SW$\Rightarrow$ Gr. I don't understand how either of the equations 2.16 or 2.17 appears, I would be happy to understand how the curvature term $F_a$ appears ...
- 853
2
votes
0
answers
67
views
Is inverse image along finite group quotient $t$-exact for the perverse $t$-structure?
Let $q: \mathbb{C}\to \mathbb{C}$ be the quotient by $\mathbb{Z}/n\mathbb{Z},$ i.e. the map taking $z\mapsto z^n$. In the accepter answer to Operations on perverse sheaves on disk the inverse image of ...
- 1,021
5
votes
1
answer
142
views
A pushout diagram of derived categories coming from an open cover of schemes
Suppose $X=U\cup V$ is the standard open cover of $X=\mathbb{P}^1$ by two affine lines. The descent theorems say that the diagram (with all arrows restriction maps)
$\require{AMScd}$
\begin{CD}
D(X) @&...
- 53
0
votes
0
answers
47
views
Reference for packing property and König property
Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
- 11
1
vote
0
answers
85
views
Hölder continuity for cocycles with respect to metrics
Let $T: X \to X$ be a uniform continuous (or Lipschitz continuous) on a compact metric space $(X, d_1)$. Assume that $Y$ is a Banach algebra and $f:X \to Y$ is a Hölder continuous with respect to the ...
- 1,011
3
votes
2
answers
280
views
A paper of Borel (in German) on compact homogeneous Kähler manifolds
I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German
Satz I: Jede zusammenhängende kompakte homogene ...
-1
votes
0
answers
43
views
What are the relations among canonical basis and dual canonical basis of the group of $A_4$?
How to construct the dual canonical basis of the $A_4$ type group from its canonical basis?
or What are the relations among canonical basis and dual canonical basis of the the $A_4$ type group?
Thank ...
- 1
5
votes
2
answers
581
views
A version of Hilbert's Nullstellensatz for real zeros
$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
- 118k
-6
votes
0
answers
47
views
Line of sight calculation - human eye [closed]
background: there is a plot of land that has a high point looking over toward the ocean/horizon downhill. A large bank of trees forms 100 feet away from the hill down toward the ocean blocking the ...
- 1
0
votes
1
answer
179
views
Atiyah sequence of a coherent sheaf
I want to show that if $X$ is a smooth complex projective variety, then analytification induces an equivalence of categories between algebraic integrable connections on $X$ and analytic integrable ...
- 361
-1
votes
0
answers
54
views
On the structure of the zeros of real polynomials of several real variables
Let $P(x_1,x_2,...,x_n)$ be a polynomial with real coefficients in the real variables $x_1,x_2,...,x_n$ that vanish on the real quadratic surface $Q(x_1,x_2,...,x_n )=0$ where $x_1,x_2,...,x_n$ are ...
1
vote
0
answers
36
views
Leibniz formula for Fulton's divided difference operators for quantum Schubert polynomials
Schubert polynomials are polynomials in the ring $\mathbb{Z}[x]$ where $x$ is an infinite set of variables indexed by the positive integers and they can be expressed in terms of "standard ...
- 2,038
1
vote
0
answers
65
views
Vanishing of chow group of 0-cycles for affine, simplicial toric varieties
Let $k$ be an algebraically closed field of characteristic zero.
Let $X$ be an affine, simplicial toric variety over $k$.
If $X$ has dimension one, then it is the affine line over the field $k$, so ...
- 531
1
vote
0
answers
66
views
When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 159
2
votes
0
answers
189
views
Some naive questions about pro-etale cohomology
Here's my dilemma: I'm trying to do some work with the pro-étale topology and would like to compute very basic instances of pro-étale cohomology to see what I can extract from a pro-étale Kummer exact ...
- 472
0
votes
0
answers
56
views
Geometric intuition behind hyper-sphere volume recurrence relation [closed]
There is a recurrence relation for calculating the volume of a hyper-sphere and a logical explanation for why it holds, as well as other explanations here on MO. Is there a geometric intuition behind ...
- 1
6
votes
4
answers
581
views
What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?
I asked this question on MSE here.
Given the length of the sides of a quadrilateral $a,b,c,d$ ( side lengths are given in order around the quadrilateral) the area of the quadrilateral is less than or ...
- 207
0
votes
0
answers
93
views
Direct image of a vector bundle defined on an open subscheme
Let $X$ be a noetherian irreducible scheme, of dimension $\geq 2$, $P\in X$ a closed point and $U=X\setminus\{P\}$. Let $\iota:U\to X$ be the inclusion morphism.
Let $E$ be a vector bundle on $U$, and ...
- 729
2
votes
2
answers
422
views
Are Chern classes always vertical?
Let $c_k \in H^{2k}(M, \mathbb{Z})$ be the $k$-th Chern class of the tangent bundle of a Hermitian manifold $M$.
Is $c_k$ necessarily vertical, i.e.
$$
c_k = \sum_{i_1,\dots, i_{k}} \alpha_{i_1 \dots ...
- 105
0
votes
1
answer
215
views
On zeros of real polynomials in two variables
Let $P(x,y)$ be a polynomial with real coefficients in two real variables $x,y$ such that the set of zeros of $P(x,y)$ is the real conic curve $Q(x,y)=0$. Will it be true that there exists a ...
4
votes
0
answers
98
views
A compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group
Is it possible to have a compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group? It seems not to be the case, but a precise argument of reference would be great!
Edit: ...
2
votes
0
answers
209
views
Representability of moduli problem of elliptic curves with complex multiplication
I'd like to know whether the moduli problem for elliptic curves with complex multiplication by a fixed imaginary quadratic number field $K$ (and with suitable level structure to be picked) is ...
- 79
4
votes
0
answers
250
views
Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?
I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$.
Recently I was pointed to Katz and Mazur's book, ...
- 33.9k
2
votes
0
answers
78
views
Smooth non-complete intersection in $(\mathbb{C}^*)^n$?
Are there examples of smooth irreducible subvarieties of $(\mathbb{C}^*)^n$ of dimension $d$ that cannot be cut out scheme theoretically by $n-d$ Laurent polynomials? If yes, how to construct them? ...
- 2,883
3
votes
0
answers
75
views
References for orbifold curves
I am looking for a good reference (if there is any) for the theory of orbifold curves from the perspective of stacks. By an orbifold curve I mean something like a $1$-dimensional irreducible Deligne-...
- 321
2
votes
0
answers
135
views
How many unit cubes are needed to 'hide' a unit cube fully in 3D?
Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...
- 5,493
5
votes
0
answers
193
views
Is it decidable whether a statement about reals (in the language of ordered rings) is constructively provable?
The language of ordered rings is a first-order language with operators for $+$, $-$, and $\cdot$, constants for $0$ and $1$, and relations for $<$, $=$ and $>$.
To decide whether such a ...
- 5,831
2
votes
0
answers
94
views
What is the correct definition of intermediate Jacobian for this singular threefold?
I am considering blow up of $\mathcal{C}\subset(\mathbb{P}^1)^3$, $X=\operatorname{Bl}_{\mathcal{C}}(\mathbb{P}^1)^3$, where $\mathcal{C}$ is a curve given by $$\{s^2u=0\}\subset\mathbb{P}^1_{s:t}\...
- 1,942
1
vote
0
answers
50
views
Parabolic (double) quantum Schubert polynomials Pieri formula
I am writing calculation software for computing structure constants of equivariant quantum Schubert polynomials and I discovered that partial flag varieties corresponding to parabolic subgroups have ...
- 2,038
2
votes
0
answers
147
views
Hodge bundles associated to a family of complex manifolds
I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem:
Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...
- 21
2
votes
1
answer
132
views
Pullback morphism of a hyperplane inclusion is zero in the derived category
Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero ...
0
votes
1
answer
106
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 345
4
votes
1
answer
173
views
Taubes' SW$\Rightarrow$ Gr
I am reading Taubes' paper on SW$\Rightarrow$ Gr and lost in some analysis, can anyone help me to see how to get equation 2.19 from equation 2.18? Is this some version of Kato for the Laplacian?
- 853