The affine-geometry tag has no usage guidance.

**4**

votes

**2**answers

55 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) := ...

**1**

vote

**0**answers

79 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

**1**answer

232 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**

vote

**0**answers

62 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 ...

**4**

votes

**0**answers

181 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**

vote

**0**answers

65 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**

vote

**0**answers

56 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

**1**answer

110 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 ...

**2**

votes

**0**answers

148 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

**1**answer

311 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

**1**answer

133 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 ...

**2**

votes

**0**answers

86 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**

votes

**0**answers

83 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

**1**answer

81 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

**0**answers

307 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 ...

**7**

votes

**2**answers

282 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 affinity?
Assuming continuity would certainly ...

**33**

votes

**1**answer

714 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

**2**answers

249 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 ...