Questions tagged [affine-geometry]

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Existence of an open set with specific properties

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 $\overline{\mathbb{F}}_p$. ...
14
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3answers
817 views

Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

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? ...
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1answer
94 views

How does the affine Desargues theorem imply the little Desargues theorem?

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
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1answer
210 views

If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?

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
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0answers
79 views

Segre's theorem in $3$ dimensions with a "twist"

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
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1answer
249 views

What are some compact Hessian manifolds?

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 ...
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0answers
23 views

On the perturbation of one vertex of an n-simplex so that point the given point on one of its face gets into its interior

I have a closed cube $Q_{0}=[0,1]^l$ and half-plane $H=\{(z_1,z_2,...,z_l) \in \mathbb{R}^l : z_1 + z_2 +...+ z_l > \alpha \}$ in Euclidean space $\mathbb{R}^l$ with $0<\alpha<1$. Consider ...
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20 views

How to prove continuity in topological group action of ${\rm{GA}}_a(X)$ on $T(X)$, to make ${\rm{GA}}(X)$ a topological group?

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 ...
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1answer
93 views

Ellipsoid in $L^p([0,1],\lambda)$ spaces?

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 ...
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1answer
245 views

Separating a lattice simplex from a lattice polytope

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 ...
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94 views

How to check if there exists a non-negative element in an affine space?

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+...
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93 views

Lie correspondence for Ind groups

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 ...
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1answer
122 views

Polynomial isometries of $\mathbb{A}^2_\mathbb{C}$

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 ...
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1answer
182 views

On a condition on ideals viwed as a Zariski open condition on co-tangent space

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 \...
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1answer
60 views

For which sets of $(n, m, k)$ does there exist an edge-labelling (using $k$ labels) on $K_n$, such that every single-labelled subgraph is $K_m$?

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 ...
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0answers
35 views

One parameter change of a section of $T^*M \otimes End(TM)$ on an affinely flat manifold

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 $...
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1answer
332 views

Covering the disk with a family of infinite total measure - the convex sequel

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). ...
2
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1answer
139 views

Intersection of a $\mathbb{Q}$-affine space with $\mathbb{Z}^n$

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 ...
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1answer
513 views

Covering the disk with a family of infinite total measure

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 ...
2
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1answer
283 views

Reference for "topological affine spaces"

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-...
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0answers
132 views

Barycenters and the axiomatic of affine geometry

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 ...
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4answers
2k views

Projective to Affine?

Perhaps a basic question... how does one study affine algebraic geometry via projective geometry? For example, suppose I have two affine varieties which I want to prove are isomorphic, would it help ...
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0answers
687 views

Theorems in affine geometry which can be proved using only dilations, generalized to metric spaces

Background: significant parts of E. Artin, Geometric algebra, Wiley-Interscience, New York, 1957 can be seen as consequences of the statement (1) "the inverse semigroup generated by dilations in ...
5
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1answer
142 views

Cohn-Vossen rigidity theorem in hyperbolic space

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 ...
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0answers
62 views

complexity of system of equations defining affine variety

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 ...
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2answers
408 views

Bieberbach theorem for compact, flat Riemannian orbifolds

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
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0answers
242 views

Flatness of modules over dual numbers

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
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1answer
469 views

Projective-invariant differential operator

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) = ...
5
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1answer
202 views

Is a symmetric, parallel (0,2)-tensor a metric?

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 ...
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1answer
475 views

How many rich directions does a set in $\mathbb F_p^2$ determine?

$\newcommand{\F}{\mathbb F}$ 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|&...
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0answers
140 views

Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains

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[...
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0answers
138 views

Factorially closed, finitely generated $k$-sub-algebra of $k[X_1,X_2,X_3]$ , where $k$ is algebraically closed field of positive characteristic

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 ...
3
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1answer
270 views

Cancellation problem for Laurent polynomial rings and power series rings

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 ...
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0answers
72 views

If a partial normalisation of an affine variety is affine?

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 ...
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3answers
504 views

Line-preserving bijection of ${\mathbb{R}}^n$ onto itself

If $f:{\mathbb{R}}^n\to{\mathbb{R}}^n$ $(n\ge2)$ is a bijection such that the image of every line is a line (continuity of $f$ not assumed), must $f$ be an affinity? Assuming continuity would ...
10
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1answer
1k views

Who first proved the fundamental theorem of projective geometry?

The following theorem is often called the fundamental theorem of projective geometry: Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of $k^n$....
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1answer
192 views

What conditions are sufficient to prove a transformation in $\mathbb{R}^2$ is affine? [closed]

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
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0answers
235 views

Almost blocking sets in $\mathbb F_q^2$

$\newcommand{\F}{{\mathbb F}}$ 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
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1answer
230 views

Automorphism group of fiber products of schemes

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 $B$ ?...
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1answer
239 views

Problem on triangles

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 ...
2
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1answer
75 views

Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$

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\...
3
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3answers
246 views

Space of halfspaces

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
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1answer
457 views

Definition of Affine Root System: What is $\alpha_{+}$?

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
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1answer
179 views

Maximin diameter of a transformed convex figure

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
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2answers
326 views

Affine spaces as algebras for an operad?

(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 ...
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2answers
946 views

When is a flow geodesic and how to construct the connection from it

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, ...
4
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1answer
255 views

Square-free sets in $\mathbb F_2^n\oplus\mathbb F_2^n$

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^...
7
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0answers
280 views

Status of an open question in Artin's "Geometric Algebra"

In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2). The ...
3
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1answer
97 views

Sets blocking every $2$-flat in $AG(n,2)$

The following may be well-known $-$ but not known to me: What is the smallest possible size of a set in ${\mathbb F}_2^n$ that blocks every $2$-flat? Here "blocks" means "have a non-empty ...
4
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2answers
357 views

How to show the two convex bodies are affinely isomorphic?

This problem comes from the response of the author of papers. Consider two convex bodies $A$ and $B$: $$A= \{X\in \mathcal{S}^4 : \operatorname{tr}(X) = 1, X\succeq 0 \}$$ $$B = \operatorname{...