All Questions
Tagged with ag.algebraic-geometry incidence-geometry
9 questions
31
votes
2
answers
1k
views
The Sylvester-Gallai theorem over $p$-adic fields
The famous Sylvester-Gallai theorem states that for any finite set $X$ of points in the plane $\mathbf{R}^2$, not all on a line, there is a line passing through exactly two points of $X$.
What ...
- 20.8k
15
votes
4
answers
966
views
Synthetic projective lines
The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and ...
- 66.8k
9
votes
2
answers
383
views
Does the Affine Pappus Axiom imply the Affine Desargues Axiom in affine planes?
I am interested in the affine version of the well-known Hessenberg's Theorem (saying that Pappian projective planes are Desarguesian).
First I introduce all necessary definitions.
Definition L. A ...
- 42k
8
votes
2
answers
972
views
Incidence geometry and matrices
Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines $s_1,\dots,...
- 13.9k
7
votes
1
answer
347
views
A corollary of the affine Desargues axiom
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
- 42k
4
votes
0
answers
102
views
Bounds on k-tuple points for intersections of hyperplanes
Suppose that $H_1$,...,$H_d$ are hyperplanes in $\mathbb P^n$ (over some field -- you can pick). For $k \geq n$, let $t_k$ denote the number of points through which there pass exactly $k$ hyperplanes....
- 41
2
votes
1
answer
153
views
Very symmetric quadrangle in $\Bbb CP^2$
Is there a quadrangle $Q \subset \Bbb CP^2$, namely $Q$ is a set of four points, such that every permutation of $Q$ can be realizad by an isometric projectivity of $\Bbb CP^2$?
Clearly the analogous ...
- 2,283
2
votes
1
answer
103
views
Is every Cartesian biaffine plane affine?
This question concerns the (synthetic) geometry of linear spaces.
Definition 1. A linear space is a pair $(P,\mathcal L)$ consisting of a set $P$ whose elements are called points and a family $\...
- 42k
2
votes
0
answers
165
views
A direct proof that every projectivity between parallel lines is affine
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
- 42k