Questions tagged [finite-geometry]
87 questions
16
votes
1 answer
How many ways to arrange $n$ points in $(\Bbb F_q)^2$ with no three collinear?
How many ways are there to arrange $n$ points in the finite field plane $(\Bbb F_q)^2$ with no three of the points collinear?
An easy upper bound is $(q^2)^n=q^{2n}$, but of course it's less than that. (Of course, if I asked the same question over…
Akiva Weinberger
- 25,412
12
votes
2 answers
How to imagine vector spaces (and projective spaces) over a finite field
So I have been learning about projective spaces for the last few hours, and I think I understand the basics pretty well, but there is an exercise, which I do not know how to solve at all. It comes down to being able to imagine and understand vector…
Jack4t3
- 643
10
votes
1 answer
groups of conics
Let $\mathcal{C}$ be a conic on the projective plane $PG(2,\mathbb{F})$ where $\mathrm{char}\mathbb{F}\neq 2$. Let $\ell$ be a line and let $N$ be a point on $\mathcal{C}\setminus \ell$. For $A,B\in \mathcal{C}$, let $L_{AB}=AB\cap \ell$. Prove that…
mandella
- 1,912
10
votes
2 answers
Is there a slick way to show that finite projective planes of $7$ points are unique up to isomorphism?
I was reading about the Fano plane, the smallest possible projective plane. After playing around with it, it seems that any projective plane of 7 points will be isomorphic to the Fano plane.
However, I've always been troubled with showing…
Dani Hobbes
- 2,795
7
votes
1 answer
Projective space definitions
My questions are as follows:
Are all these different definitions of projective space equivalent? For example, Bezout's theorem holds under all 4 definitions (with an appropriate change in terminology)?
Are the projective topological spaces…
rationalbeing
- 1,267
7
votes
3 answers
Who conjectured that there are only finitely many biplanes, and why?
This question on MathOverflow motivates me to ask what the reasoning is behind the conjecture that there are only finitely many biplanes. More generally, it has been conjectured that for fixed $\lambda>1$ there are only finitely many triples…
Will Orrick
- 19,218
6
votes
0 answers
Is a perfect game of Set always possible?
For anyone not familiar with the game of Set, I'll refer you to the description on this question.
My question is this: The game ends when there are no more cards remaining in the deck and there are no sets visible on the table. I will define a…
Darrel Hoffman
- 977
6
votes
2 answers
Primitive roots of unity occuring as eigenvalues of a product
I am currently trying to understand the proof of Benson's Lemma (1.9.1) in Generalized Quadrangles by Payne and Thas.
Background
We have two $k × k$ matrices $Q$ and $M$. We want to determine some formula for $\operatorname{tr}(QM)$. The following…
Santana Afton
- 6,978
6
votes
1 answer
Connection between linearly independent vectors and projective points in general position
I'm trying to understand the connection between the notions of linear independence and general position. I have no background in geometry, so first I'll start with what I know and then I'll pose specific questions, please bear with me and correct me…
geo909
- 542
6
votes
2 answers
How large can a set of pairwise disjoint 2-(7,3,1) designs (Fano planes) be?
As wikipedia defines well, the Fano plane is a small symmetric block design, specifically a 2-(7,3,1)-design. The points of the design are the points of the plane, and the blocks of the design are the lines of the plane.
Now what I'm trying to find…
Niloo
- 452
5
votes
2 answers
About Fano plane symmetries
I have been looking for realizations of order $21$ metacyclic group. I asked about this yesterday and learned some very good information about it Realization of the metacyclic group of order 21
I was also experimenting with the Fano plane and…
user581023
5
votes
0 answers
Translating and inflating a set of $k$-dimensional subspaces of $\mathbb F_p^n$ to form a cover by affine hyperplanes?
Fix a prime number $p$ and consider the affine space $V = \mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, \ldots, V_n \subseteq V$ of dimension $k$, and take $v_i \notin V_i$. Do there always exist subspaces $W_i \supseteq V_i$ of…
Bart Michels
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5
votes
2 answers
How to interpret a line equation in 4-point geometry (affine plane of order 2).
I am currently reading "Basic Notions of Algebra" by Igor Shafarevich. In the first chapter example of a coordinatization of 4-point geometry is given.
Set of axioms:
Through any two distinct points there is one and only one line.
Given any line…
4
votes
0 answers
State of Research on Projective Planes of Order 11
It's been 3 decades since it was shown that there is no projective plane of order 10. I can find lots of commentary and references on the state of play on the attempt to show there is no projective plane of order 12. It seems to be an open…
C Monsour
- 8,476
4
votes
1 answer
Conjugacy of Singer cyclic groups in $\mathrm{P\Gamma L}$
Motivation
This is kind of a follow-up to this question on conjugacy of Singer cyclic groups in GL.
The "original" definition of a Singer cycle is not in the GL, but the following slightly different geometric setting:
Definition
A cyclic group…
azimut
- 24,316