All Questions
32,947
questions
1
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0
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11
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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,...
- 494
1
vote
0
answers
28
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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, ...
- 985
2
votes
0
answers
37
views
Infinitesimal neighborhood and Ext group
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}=H^p(\wedge^q N_{Y/X})\...
- 131
2
votes
1
answer
72
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}$).
...
- 167
1
vote
1
answer
76
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
1
vote
1
answer
47
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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$ ...
- 609
2
votes
0
answers
30
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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
41
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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}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
- 1,417
1
vote
0
answers
55
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
60
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
2
votes
1
answer
75
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) @&...
- 21
0
votes
0
answers
44
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
57
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
261
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
41
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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
557
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
46
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
172
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
53
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
33
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
60
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
64
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
175
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
55
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
490
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
87
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
407
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
211
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
96
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
203
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
242
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
68
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
72
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
132
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
191
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 ...
- 5,821
2
votes
0
answers
92
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
142
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
131
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
103
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 ...
- 341
4
votes
1
answer
171
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
4
votes
0
answers
156
views
Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,109
2
votes
0
answers
122
views
GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,681
4
votes
1
answer
215
views
Compactification of rigid-analytic varieties
Is it true that any separated quasi-compact rigid-analytic variety embeds into a proper one?
For my purpose, the base field is a $p$-adic number field.
I have seen Huber's universal compactification ...
- 91
1
vote
0
answers
99
views
Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension
Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
- 265
2
votes
1
answer
149
views
Blowup formula for a morphism
Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...
3
votes
1
answer
141
views
Product of low dimensional Hausdorff measures
Let $\mathcal{H}^n$ and $\mathcal{H}^m$ be Hausdorff measures on $\mathbb{R}^n$ and $\mathbb{R}^m$. We know that the product measure $\mathcal{H}^n\otimes \mathcal{H}^m$ is the Hausdorff measure $\...
- 71
3
votes
0
answers
174
views
Is pullback map on sheaf cohomology injective for surjective morphisms?
Consider a surjective map $f\colon X\to Y$ of smooth projective varieties. It is well known (see e.g. Voisin's Hodge theory I, Lemma 7.28) that the map $H^i(Y,\mathbb Q)\to H^i(X,\mathbb Q)$ is ...
- 2,235
1
vote
0
answers
75
views
A homogeneous manifold that does not admit an equivariant Riemannian metric?
Let $M = G/H$ be a homogeneous space, where $G$ is a Lie group and $H$ is a closed Lie subgroup. Can it happen that $M$ does not admit an invariant Riemannian metric?
- 1,144
1
vote
0
answers
114
views
Solution formula in an explicit equation over $\mathbb{F}_p^3$
I'm looking into a formula involving prime numbers $p \geq 7$ and an equation's solutions. The equation in question is:
$$z^2 = (x^2 - 4x)(y^2 - 4y)((x + 1 - y)^2 - 4x),$$
where $(x,y,z)\in \mathbb{F}...
- 71