1
vote
0answers
47 views

Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
-2
votes
0answers
24 views

Euclidean geometry problems for thirs year students [on hold]

Can you suggest some books with exercises related to Euclid's Elements, or to Euclidean Geometry, as an aid to an undergraduate course on Euclidean Geometry and its history? I need exercises that ...
2
votes
1answer
72 views

Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban "The Iwasawa main conjecture for $GL_2$"? I am interested in partucular in the case of elliptic ...
2
votes
0answers
52 views

Invariant definition of the space of symbols on a vector bundle

Normally a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates \begin{equation} \sup_{x \in U, \xi \in ...
4
votes
0answers
122 views

Counterexamples to Elkik's theorem in the non-Noetherian case

Elkik in Solutions d'equations a coefficients dans un anneu Henselian, Theorem 7 proves that: Let $A$ be a Noetherian ring that is Henselian with respect to a principal ideal $(a)$. That is, if ...
6
votes
4answers
349 views

A metric space of geometric shapes

My research involves geometric shapes in $R^2$, and I need a metric with several properties such as: Families of similar shapes, such as squares, are closed in this metric. Also more general ...
1
vote
2answers
61 views

Complementary integrable vector fields

Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric ...
2
votes
0answers
67 views

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...
1
vote
1answer
83 views

Estimating the volume of a union of balls

Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$. If I want to estimate $$ \frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1} $$ where ...
2
votes
0answers
75 views

Focal points for the exponential map and Jacobi fields

It is known that in a Riemannian manifold $(M,g)$, if there is a closed geodesic and a non-zero, periodic, non-constant Jacobi field along it, then M has a focal point. Is the converse true? That is ...
1
vote
1answer
111 views

Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...
0
votes
0answers
115 views

Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
0
votes
0answers
7 views

Minimum height of convex area, with minimum area dependant on number of fixed length sides [migrated]

I've come across a problem while coding that can be solved simply, to an adequate standard, but I'm sure has a very interesting perfect solution. The problem is as follows: Given N lines of ...
0
votes
0answers
90 views

Galois descent of stacks

Suppose $X$ is a algebraic stack over a field $F$. Consider a Galois finite extension $F'$ of $F$. Let $X_{F'}:=X\times_F F'$ be the base extension. Suppose $X_{F'}$ is isomorphic to an algebraic ...
0
votes
1answer
71 views

linear section of codimension $k+1$ of a variety of dimension $k$

Let $X$ be a variety of dimension $k$ and degree $d$. If $L$ is a linear subspace of codimension $k+1$ such that $|L\cap X|$ consists of a finite number of points, is there a way to find the maximum ...
0
votes
1answer
55 views

Can we always solve this equation in the space of Hermitian structures on a complex vector bundle?

Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$. (I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2) Assume we ...
1
vote
1answer
49 views

holomorphic vector fields on CP^2 blow ups

On $X=CP^2\#k{(-CP^2)}$ in $k$ generic points, Let $h^i=dim H^i(T^{1,0}X)$, for $i\ge 0$. First we know $h^i=0$ for $i\ge 2$. By Riemann-Roch formula, I obtain that $h^0-h^1=8-2k$. Would some one be ...
0
votes
0answers
28 views

(semi)stability of a net of quadrics and a certain criterion

Let $V:=H^0(\mathbb{P}^4,\mathcal{O}(1))$, $W:=H^0(\mathbb{P}^4,\mathcal{O}(2))$ and $Q_i(i=1,2,3)$ be quadric hypersurfaces in $\mathbb{P}^4$. To a net of quadrics $\Lambda=(Q_1,Q_2,Q_3)$, we ...
0
votes
1answer
101 views

Degree of a Segre variety (without Hilbert polynomial)

Algebraic Geometry is not my topic of research and I am having troubles in understanding the following: In Harris book, Algebraic Geometry, a firs course, Example 18.15, there is a proof of the ...
0
votes
1answer
46 views

Global sections in negative degree line bundles over singular curves

Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth ...
2
votes
0answers
105 views

$S^{3}$-valued harmonic analysis

Edit: Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider $$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid ...
8
votes
1answer
748 views

What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...
4
votes
0answers
187 views
+50

What arrangement of unit cubes minimizes surface area?

Question A. How does one arrange $n$ unit cubes to minimize surface area? Question B. How does one arrange $n$ unit cubes to form a rectangular prism of minimal surface area? Various curricular ...
-3
votes
3answers
96 views

if V(f) is irreducible, then how to show that the polynomial f itself is irreducible? [on hold]

V(f) is the zero locus of the polynomial f in the polynomial ring k[x1, x2, ..., xn] with k an algebraically closed field. If V(f) is irreducible, then how to show that 'f' is irreducible?
0
votes
2answers
166 views

Weil restriction

I've already asked a similar question in SE, without success, so I've decided to post here a more general version of my question. Let $f: Y \to X$ be a finite etale morphism of smooth proper ...
3
votes
1answer
116 views

Rational homogenous spaces and symmetric spaces

What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian ...
3
votes
0answers
44 views

Polynomials connected with Gale's condition and cyclic polytopes

I was reading about cyclic polytopes and (I think naturally) became interested in polynomials $f(X) \in \mathbb{R}[X]$ which have a factorization of the form $f(X)=g(X)g(X+1)$ for some $g(X) \in ...
1
vote
0answers
45 views

symmetric theta structures and arithmetic subgroups

A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface. ...
2
votes
2answers
103 views

Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map $$ N = \begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & \\ & & 0 ...
6
votes
2answers
293 views

Twist in identification with singular cohomology

Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...
3
votes
1answer
69 views

$\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$. Moreover, let $M \colon \mathbb{R}^n \to ...
0
votes
0answers
109 views

Hermitian metric on $f_*(\Omega_{X/X_{can}}^{n-k})$

Let $X$ be an n- dimensional algebraic manifold . Suppose that its canonical line bundle $K_X$ is semi-positive and $0<k=Kod(X)<n $ . Let $f: X\to X_{can}\subset \mathbb CP^N$ Here $X_{can}$ ...
2
votes
0answers
45 views

How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base [migrated]

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
6
votes
1answer
209 views

Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
2
votes
0answers
97 views

A question on resolution of singularities

I am wondering if it could be possible in particular cases to resolve a singularity of dimension $n$ by blowing-up a locus of dimension smaller than $n$. For instance consider a cubic surface ...
1
vote
1answer
58 views

Real solutions for systems of monomial equations

I have a $K$ equations of the form $x_1^{a_{i1}} \cdots x_n^{a_{in}}=c_i$ where $a_{ij}$ are non-negative integer constants and $c_i$ are real constants -- i.e. each equation is a monomial in $n$ ...
3
votes
0answers
97 views

Non-linearly isomorphic non-equivalent $G-$representations?

Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow ...
5
votes
1answer
274 views

What is the current state of the crystalline analogue of the Weil conjectures?

In "F-isocrystals on open varieties results and conjectures" Faltings says: "Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil ...
0
votes
0answers
171 views

Which of the Mochizuki's works are the most closely related to elliptic curves?

I'm very much interest about algebraic geometry and number theory along with cryptography, but I have a special interest about the elliptic curves. I have heard a lot of interesting things about ...
18
votes
2answers
805 views

A question about “Zariski dense” arguments

This question is a little basic, but I think it is consistent with the goals of MO. My question is about a certain type of argument in algebraic geometry which exploits the abundance of dense sets ...
0
votes
1answer
51 views

equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
7
votes
2answers
288 views

Is there a hyperplane avoiding two independent sets?

Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
1
vote
1answer
366 views

Standard conjectures on positive characteristic

In this MO answer of M. Bondarko, he says: "the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..." and in Remarks on Grothendieck's ...
7
votes
1answer
202 views
+150

Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$. On the other ...
3
votes
5answers
193 views

Approaching convex and discrete geometry from other disciplines

I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning ...
2
votes
1answer
105 views

Nearby cycles and specialisation - properties

I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
1
vote
2answers
137 views

Plucker embedding and tautological/universal quotient bundle

Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the ...
3
votes
0answers
178 views

A lifting problem

Let $E\overset{\pi'}{\longrightarrow} B'$ and $E\overset{\pi}{\longrightarrow} B$ be vector bundles. For $i=0,1$, let $f_i$ be a fiber-preserving open embeddings of $\pi'$ into $\pi$, with $g_i$ the ...
1
vote
1answer
38 views

Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton

I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations ...
2
votes
0answers
46 views

Linear transformation of a polyhedral cone

Let $C$ be a polyhedral cone in $\mathbb{R}^m$ with H- and V-representations $C = \{x : A x \le 0 \} = \{R y : y \ge 0\}.$ The pair $(A,R)$ is referred to as a double description (DD) pair of the ...