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Questions tagged [affine-geometry]

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5
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1answer
100 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 ...
1
vote
0answers
94 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[...
1
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0answers
110 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
votes
1answer
139 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 ...
1
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0answers
46 views

Factorially closed, finitely generated $k$-sub-algebra $A$ of $k[X_1,…,X_n]$, where $n>3$, $k$ is algebraically closed of char $0$, $trdeg_k A=n-1$

Let $S$ be a sub-ring of a commuttaive ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S \implies a,b \in S$. My question is : Let $k$ be an algebraically ...
8
votes
1answer
367 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
69 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 ...
0
votes
1answer
149 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 ...
8
votes
0answers
181 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
vote
1answer
146 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 $B$ ?...
3
votes
3answers
234 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
votes
1answer
64 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\...
1
vote
1answer
193 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
votes
1answer
174 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
votes
2answers
295 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 ...
11
votes
3answers
432 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? ...
12
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2answers
630 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, ...
6
votes
1answer
228 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 ...
4
votes
1answer
245 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^...
6
votes
0answers
235 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
votes
1answer
82 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
votes
2answers
238 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{...
0
votes
1answer
281 views

Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true. Since posting the question, ...
5
votes
1answer
267 views

What is meant by a Lie group acting by affine transformations?

This question is a continuation of this one, where I did not receive a complete answer so am moving to MO. I am trying to understand a paper by JP Szaro that was referred to there, and in particular, ...
1
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0answers
80 views

Characterizing normal vectors of affinely independent subsets of the hypercube

Suppose we are looking at the hypercube in $\mathbb{R}^n$. I tend to the think of the cube with vertex coordinates 0 or 1, but maybe this is easier for the $\pm1$ cube. Now suppose we have an affine ...
5
votes
1answer
158 views

Affinely flat structures. How many different ones on the same manifold?

Let $M$ be a smooth manifold endowed with $\nabla$ a flat torsion-free connection. Let $\operatorname{Diff}(M)$ be the group of smooth diffeomorphisms of $M.$ Obviously if $ \phi \in \operatorname{...
4
votes
2answers
102 views

Affine hull of a set of non-negative matrices with fixed row-sums

Fix any non-negative matrix $M \in \mathbb{R}_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column. Further, fix any positive vector $r \in \mathbb{R}_{> 0}^m$. With $nz(M) := \{(i,j)...
1
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0answers
104 views

What is this 2-form on a Lagrangian torus fibration?

Suppose we are given a regular $2n$-dimensional Lagrangian fibration $\pi : (M,\omega) \to B$ with connected, compact fibers. Then it is well-known (Arnold-Liouville) that each fibre is a Lagrangian ...
5
votes
1answer
293 views

$(n-2)$-blocking sets in $AG(n,2)$

Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$. I have seen a lot work related to minimal $(n-1)$-blockings set. ...
1
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0answers
111 views

Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb R^2$...
5
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0answers
219 views

Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...
1
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0answers
68 views

Uniform bounds on affine invariants of a family of manifolds

Suppose $\{M_i\}$ is a family of $C^{\infty}$ smooth, strictly convex hypersurfaces in $\mathbb{R}^n$ such that $B_r\subseteq M_i\subseteq B_R$, where $B_r,B_R$ are balls centred at the origin of ...
1
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0answers
127 views

A questions related to the Markus conjecture for special affine manifolds

An affine manifold $M$ is called special if there is a parallel volume form $\omega$ on $M$, and a nowhere vanishing vector field $\mathcal{V}.$ Here we need to point out that any affine manifold of ...
1
vote
1answer
146 views

Uniqueness affine curvature

Let $\gamma_1,\gamma_2: \mathbb{S}^1 \to \mathbb{R}^2$ be two smooth, closed, convex curves that their (special)affine curvature, $\mu_1,\mu_2$ are equal, that is $\mu_1(\theta)=\mu_2(\theta)$, for ...
9
votes
1answer
850 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 $k^n$....
3
votes
0answers
207 views

Parallel Ricci condition - Status report and bibliography

First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...
8
votes
1answer
348 views

Existence of certain “nondegenerate” function and manifold topology

Let $M$ be a smooth manifold without boundary, not necessarily compact. Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...
1
vote
1answer
153 views

Affine differential geometry. Is Calabi's hypersurface isotropic?

I am in the framework of (equi)affine differential geometry. Let $E$ be a centro-equiaffine space, that is a real vector space of dimension $n$, together with the special linear group $SL_n(R)$. Let $...
3
votes
0answers
115 views

Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex if its restriction to each line is. An affine ...
0
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0answers
132 views

Compactification of the affine space with a del Pezzo surface

Is there some nice compactification of the affine space with a del Pezzo surface of degree $\le 4$ (for example with a cubic surface) ? More precisely, I would like a projective algebraic variety $X$ ...
0
votes
1answer
137 views

Is there a way to make MAGMA work with surfaces over weighted projective spaces?

Is there a way to use MAGMA to study surfaces defined over a weighted projective space (by "study" I mean computing e.g. invariants (e.g. $p_a$, $p_g$), singularities, etc)? For example, I was trying, ...
1
vote
0answers
574 views

Does the coordinate ring of affine variety admit a structure of infinite dimensional variety?

We work in the category of algebraic varieties over some algebraically closed field $k$. By infinite dimensional variety I mean a filtration: $$ V_0\subset V_1\subset V_2\subset\ldots $$ where each $...
10
votes
3answers
456 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 ...
36
votes
1answer
1k views

Is an affine fibration over an affine space necessarily trivial?

Let $X$ be an algebraic variety over an alg. closed field with zero char. and let $f:X\to \mathbb{A}^n$ be a smooth surjective morphism, such that all fibers (at closed points) are isomorphic to $\...
5
votes
2answers
270 views

Diameter-area ratio for affine tranformations.

Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$? I found only one reference, to "Über einige ...