# Questions tagged [affine-geometry]

The affine-geometry tag has no usage guidance.

The affine-geometry tag has no usage guidance.

77
questions

4
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1
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200
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Given any linear space $L$ over an ordered field $F$, consider the equiproportion relation $${\sim}=\{((x,y,z),(a,b,c))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\;(y{-}x=t(z{-}x)\wedge b{-}a=t(c{-...

4
votes

1
answer

191
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Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\text{M}_n(\mathbb{F}_q)$ be the vector space of $n \times n$ matrices over $\mathbb{F}_q$, let $\text{GL}_n(\mathbb{F}_q)$ be the group of $...

3
votes

0
answers

45
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This is a 2-person game.
Let $\ P\ $ be any arbitrary projective space (of any dimension $\ \ge2$ and any cardinality, etc., but typically, let it be a finite plane over a field). Let $\ S_0\subseteq ...

-1
votes

1
answer

164
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Can a regular quasi affine variety (i.e. open subscheme of an affine variety) contain a (possibly singular) projective curve?

0
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0
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47
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I am using a 3D Affine Transformation matrix of the form $T(x) = RGS (x-c) + t + c$, with:
$R= R_x R_y R_z$ the rotation matrix,
$G$ the shearing matrix,
$S$ the scaling matrix,
$c$ the center of ...

2
votes

1
answer

234
views

Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let
$$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$
be a local parametrization of $M$.
...

3
votes

0
answers

30
views

Let $\mathcal{A}$ be an axiomatic affine plane. First let $\mathcal{A}$ be finite.
Suppose that the automorphism group of $\mathcal{A}$ acts transitively on nonincident point-line pairs (that is, on ...

5
votes

0
answers

110
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Consider the intersection of an $n$-dimensional hybercube and an $m$-dimensional flat (affine subspace) which contains the diagonal of the hypercube. This is a convex polytope. What is the maximal ...

1
vote

0
answers

37
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I am currently reading Gerard Laumon's Faisceaux caractères, which details various constructions and properties of character sheaves on a connected reductive group $G$ over $\overline{\mathbb{F}}_p$. ...

2
votes

1
answer

143
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Let $A$ be an (abstract) affine plane. We call $A$ a translation plane if the group of translations acts transitively on the set of points (Axiom 4a in Artin's book "Geometric algebra").
...

2
votes

1
answer

218
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This is a cross-post.
Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric.
Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...

2
votes

0
answers

84
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As I understand, there is a $3$-dimensional analogue of Segre's theorem stating that the maximum size of a set in ${\bf F}_q^3$ ($q$ odd) with no three points collinear is $q^2+1$. I am trying to ...

7
votes

1
answer

402
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In case this is too general, here is a more specific question.
Is there a hyperbolic threefold which admits a Hessian metric (hyperbolic or otherwise)?
Background
A Hessian manifold is a Riemannian ...

1
vote

0
answers

23
views

The question comes from the following paragraph of a text on geometry in the context of affine geometry (Marcel Berger et al., "Geometry I", P56-57):
2.7.1.3. If we don't want to resort to ...

0
votes

1
answer

105
views

Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure. These are Banach spaces for $p\ge1$ (of course). It is well known that for $ 1\leq p < q \leq +\infty$ we ...

1
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0
answers

107
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Suppose that a $n$-dimensional affine space is defined with vectors $e_i=(0,\dots,1,\dots,0), i=1,\dots,n$ (the $1$ stands in the $i$-th place), i.e.
$$\mathcal{A}=\left\{\alpha_1e_1+\alpha_2e_2+\dots+...

8
votes

1
answer

259
views

Let $P\subset\mathbb{R}^n$ be a convex lattice polytope.
Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex ...

2
votes

0
answers

101
views

I have the following question (with a negative answer in general).
Consider an Ind algebraic group (e.g. a group object in category of ind varieties) as defined by Shafrevich. It was assumed for some ...

1
vote

1
answer

128
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I have the following question, which I'm sure must be explored somewhere.
Consider a group of polynomial automorphisms of $\mathbb{A}^2_\mathbb{C}$ preserving a standard hermitian metric. Is there any ...

3
votes

1
answer

63
views

Or, equivalently - for what sets of $(n, m, k)$ is it possible, for a group* of $n$ people, to arrange $k$ days of "meetings", such that every day the group is split into subgroups of $m$ people, and ...

1
vote

1
answer

187
views

Let $(R, \mathfrak m,k)$ be a Noetherian local ring such that the residue field $k$ is infinite. Let $n=\mu(\mathfrak m)$. Then $n=\dim_k(\mathfrak m/\mathfrak m^2)$ . By fixing $x_1,...,x_n \in \...

1
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0
answers

37
views

Let $\nabla$ be a flat symmetric connection in the tangent bundle of a smooth manifold $M.$ Let $A$ be a global section of $T^*M \otimes End(TM).$ Let $\phi:B(0,1) \to M$ be a local affine chart on $...

2
votes

1
answer

149
views

Let $E$, a $\mathbb{Q}$-affine space of arbitrary dimension included in $\mathbb{Q}^n$. Is it possible to check efficiently if $E \cap \mathbb{Z}^n$ is empty or not?
If is an hard problem could give ...

12
votes

1
answer

341
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Let $(U_n)_n$ be an arbitrary sequence of open convex subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). ...

17
votes

1
answer

533
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Let $(U_n)_n$ be an arbitrary sequence of open subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). Does ...

3
votes

0
answers

155
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There are several ways to define an affine space,
either by starting from a transitive action of a vector
space on a set of points, or listing sets of axioms
related to parallelism in the spirit of ...

5
votes

1
answer

155
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There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf
Any isometry between two closed smooth convex surfaces in ...

1
vote

0
answers

64
views

Say you have an affine variety $X$ in $n$-dimensional affine space. (You can even assume we are over $\mathbb{C}$, but I believe the nature of my question is algebraic).
I want to bound from above ...

2
votes

1
answer

327
views

I am wondering if there is a topological version of affine spaces as a topological space along with a free transitive (continuous) action of a topological vector space on it?
Here is a notion so-...

9
votes

2
answers

480
views

In his thesis, Bieberbach solved Hilbert 18 problem and
proved that any compact, flat Riemannian manifold is a
quotient of a torus. I need a reference to an orbifold version
of this result: any ...

2
votes

0
answers

296
views

Let $X$ be a smooth, affine complex surface, and $M$ be a coherent $\mathcal{O}_X$-module. Denote by $D:=\mbox{Spec}(\mathbb{C}[t]/(t^2))$ and $X_D:=X \times_{\mathbb{C}} D$, the trivial deformation ...

14
votes

1
answer

494
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This question was originally asked on Math StackExchange.
Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that
\begin{align*}
&T(g) = ...

6
votes

1
answer

263
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I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...

1
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0
answers

155
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Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[...

1
vote

0
answers

144
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Let $S$ be a sub-ring of a commutative ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S\setminus \{0\} \implies a,b \in S$.
Let $k$ be an algebraically ...

4
votes

1
answer

368
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Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...

10
votes

1
answer

499
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$\newcommand{\F}{\mathbb F}$
A subset $P$ of the affine plane $\F_p^2$ is said to determine a direction if there is a line in this direction containing at least two points of $P$.
A set of size $|P|&...

1
vote

0
answers

72
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Let $X$ be an affine variety. Let $x$ be an isolated singular point of $X$. Let $U$ is an affine neighbourhood of $x$ such that $U\setminus \{x\}$ is smooth. Let $\pi:\tilde{U}\to U$ be the ...

0
votes

1
answer

210
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What conditions are sufficient to prove a transformation in $\mathbb{R}^2$ is affine?
I currently have shown that the transformation is bijective, points and lines are preserved and also the ...

9
votes

0
answers

240
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$\newcommand{\F}{{\mathbb F}}$
Let $q$ be an odd prime power. A blocking set in the affine plane $\F_q^2$ is a set blocking (meeting) every line.
A union of two non-parallel lines is a blocking set ...

1
vote

1
answer

250
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Let $A \mapsto S$ and $B \mapsto S$ be two schemes over the scheme $S$. Is there a connection between the automorphism group of the scheme $A \otimes_{S} B$ and the automorphism groups of $A$ and $B$ ?...

3
votes

3
answers

250
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Consider a real vector space $V$ with dimension $n$ (say $V=\mathbb{R}^n$). The construction I'm describing is similar to the construction of the projective space. Instead of the space of lines, it is ...

2
votes

1
answer

76
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I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\...

2
votes

1
answer

506
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I was beginning to read Bruhat and Tits article Groupes Reductifs sur un Corps Locale and was confused on a point in the beginning of the section on affine root systems.
$\mathbf{A}$ is a finite ...

4
votes

1
answer

181
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You are given a compact convex figure in the plane, $C$.
Your goal is to transform $C$ to a different figure $C'$ with the smallest possible diameter, as long as:
The trasformation from $C$ to $C'$ ...

10
votes

2
answers

336
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(at.algebraic-topology because I don't know who else thinks about operads)
Let $A$ be an affine space, i.e. a torsor over (the abelian group underlying) a vector space $V$ over a field $K$. Then for ...

14
votes

3
answers

886
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Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...

15
votes

2
answers

1k
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Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following:
If $\gamma:[t_0,t_1]\rightarrow M$ is a smooth curve and is a geodesic, ...

6
votes

1
answer

242
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Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l_1,l_2,l_3$ the squares of the lengths of the sides of $T$ and $l_1^t,l_2^t,l_3^t$ the squares of the lengths of the ...

4
votes

1
answer

260
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A square in $\mathbb F_2^n\oplus\mathbb F_2^n$ is a quadruple of the form
$$ (u,v)+\{(0,0),(0,d),(d,0),(d,d)\},\quad u,v,d\in\mathbb F_2^n,\ d\ne 0. $$
A set $A\subset\mathbb F_2^n\oplus\mathbb F_2^...