Incidences and projective geometries

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 2 June 2013

Incidence

An incidence geometry is a triple $(P,L,I)$ where $P$ and $L$ are sets and $I\subseteq P\times L\text{.}$

$$\begin{array}{ccccc}I& \subseteq & P\times L& \stackrel{{pr}_1}{⟶}& P\\ & & \stackrel{{pr}_2}{}\downarrow \phantom{{pr}_2}\\ & & L\end{array}$$

A point $p\in P$ is contained in a line $\ell \in L$ if $(p,\ell )\in I\text{.}$ A set of points $S\subseteq P$ is collinear if there exists $\ell \in L$ such that if $p\in S$ then $(p,\ell )\in I\text{.}$

Often it is convenient to

$$\text{identify}\hspace{0.17em}\ell \in L\hspace{0.17em}\text{with the set of points}\hspace{0.17em}{pr}_1\left({pr}_2^{-1}\left(\ell \right)\right)\text{.}$$

Subspaces

Assume that $(P,L,I)$ is an incidence geometry such that

$$\begin{array}{c}\text{if}\hspace{0.17em}{p}_1,p_2\in P\hspace{0.17em}\text{and}\hspace{0.17em}{p}_1\ne p_2\hspace{0.17em}\text{then}\\ \text{there exists a unique}\hspace{0.17em}\ell \in L\hspace{0.17em}\text{with}\hspace{0.17em}(p_1,\ell )\in I\hspace{0.17em}\text{and}\hspace{0.17em}(p_2,\ell )\in I\text{.}\end{array}$$

The line $\ell =\ell (p_1,p_2)$ containing $p_1$ and $p_2$ is the line connecting $p_1$ and $p_2\text{.}$

A subspace is a subset $S\subseteq P$ such that

$$\text{if}\hspace{0.17em}{p}_1,p_2\in S\hspace{0.17em}\text{then}\hspace{0.17em}{pr}_1\left({pr}_2^{-1}\left(\ell (p_1,p_2)\right)\right)\subseteq S\text{.}$$

A subspace is $S\subseteq P$ which contains any line connecting two of its points.

Lattices

Let $ℒ$ be a partially ordered set and let $x,y\in ℒ\text{.}$ The join, or supremum, or least upper bound of $x$ and $y$ is

$$x\vee y=\text{sup}\{x,y\}\hspace{0.17em}\text{in}\hspace{0.17em}ℒ\hspace{0.17em}\text{such that}$$

(a)	$\text{sup}\{x,y\}\ge x\hspace{0.17em}$ and $\text{sup}\{x,y\}\ge y,$ and
(b)	If $z\in ℒ$ and $z\ge x$ and $z\ge y$ then $z\ge \text{sup}\{x,y\}\text{.}$

The meet, or infinum, or greatest lower bound, of $x$ and $y$ is

$$x\wedge y=\text{inf}\{x,y\}\hspace{0.17em}\text{in}\hspace{0.17em}ℒ\hspace{0.17em}\text{such that}$$

(a)	$\text{inf}\{x,y\}\le x\hspace{0.17em}$ and $\text{inf}\{x,y\}\le y,$ and
(b)	If $z\in ℒ$ and $z\le x$ and $z\le y$ then $z\le \text{inf}\{x,y\}\text{.}$

A lattice is a partially ordered set $ℒ$ such that

$$\text{if}\hspace{0.17em}x,y\in ℒ\hspace{0.17em}\text{then}\hspace{0.17em}x\vee y\hspace{0.17em}\text{and}\hspace{0.17em}x\wedge y\hspace{0.17em}\text{exist in}\hspace{0.17em}ℒ\text{.}$$

A modular lattice is a lattice $ℒ$ such that

$$\begin{array}{c}\text{if}\hspace{0.17em}x,z\in ℒ\hspace{0.17em}\text{and}\hspace{0.17em}x\le z\\ \text{then}\hspace{0.17em}x\vee (y\wedge z)=(x\vee y)\wedge z\text{.}\end{array}$$

Projective lattices

Let $ℒ$ be a finite lattice with a unique minimal element 0 and a unique maximal element 1.

An atom is $a\in ℒ$ such that there does not exist $a\prime \in ℒ$ with $0<a\prime <a\text{.}$

An atomic lattice is a lattice $ℒ$ such that every element is a join of atoms.

A maximal chain is a maximal length sequence $0<a_1<a_2<\dots <a_{\ell }<1$ in $ℒ\text{.}$

A lattice $ℒ$ is ranked if all maximal chains in $ℒ$ have the same length.

Let $ℒ$ be a ranked lattice and let $a\in ℒ\text{.}$ The rank of $a$ is $i$ if there exists a maximal chain

$$0<a_1<a_2<\dots <a_{\ell }<1\text{with}{a}_i=a\text{.}\text{Write}\text{rank}\left(a\right)=?$$

A projective lattice is an atomic ranked modular lattice such that

$$\text{if}\hspace{0.17em}x,y\in ℒ\hspace{0.17em}\text{then}\hspace{0.17em}\text{rank}(x\vee y)+\text{rank}(x\wedge y)=\text{rank}\left(x\right)+\text{rank}\left(y\right)\text{.}$$

A projective geometry is an incidence $(P,L,I)$

$$\begin{array}{ccccc}I& \subseteq & P\times L& \stackrel{{pr}_1}{⟶}& P\\ & & \stackrel{{pr}_2}{}\downarrow \phantom{{pr}_2}\\ & & L\end{array}$$

such that

(a)	If $p_1,p_2\in P$ and $p_1\ne p_2$ then there exists a unique line $\ell (p_1,p_2)\in L$ containing $p_1$ and $p_2,$
(b)	If $p_1,p_2,p_3\in P$ are noncollinear and $\ell$ is a line intersecting $\ell (p_1,p_3)$ and $\ell (p_2,p_3)$ then there exists $p_6\in P$ contained in $\ell$ and $\ell (p_1,p_2)\text{.}$ $$p1 p2 p3 p6 ℓ(p1,p2) ℓ(p1,p3) ℓ(p2,p3) ℓ$$
(c)	Any line contains at least 3 points
(d)	There exist 3 noncollinear points in $P$
(e)	Any increasing sequence of subspaces has finite length.

Theorem Let $ℒ$ be the subspace lattice of $(P,L,I)\text{.}$ Then

$$\begin{array}{ccc}\left\{\text{projective geometries}\right\}& ⟷& \left\{\text{projective lattices}\right\}\\ (P,L,I)& ⟼& ℒ\end{array}$$

is a bijection.

Automorphisms

An automorphism of $(P,L,I)$ is

$$g\in \text{Sym}\left(P\right)\times \text{Sym}\left(L\right)\hspace{0.17em}\text{such that}\hspace{0.17em}gI=I\text{.}$$

Hence, an automorphism of $(P,L,I)$ is

$$g\in (\text{Sym}\left(P\right)\times \text{Sym}\left(L\right))\cap \text{Sym}\left(I\right)\text{.}$$

If $G$ is the automorphism group of $(P,L,I)$ then

$$\begin{array}{ccccc}I& \subseteq & P\times L& \stackrel{{pr}_2}{⟶}& P\\ & & \stackrel{{pr}_1}{}\downarrow \phantom{{pr}_2}\\ & & L\end{array}\text{is}\hspace{0.17em}G\text{-equivariant.}$$

A homology is a matrix $g$ conjugate to

$$\left(\begin{array}{c}a\\ & 1\\ & & \ddots \\ & & & 1\end{array}\right),$$

i.e. $g$ is semisimple and fixes a hyperplane.

An elation is a matrix $g$ conjugate to

$$\left(\begin{array}{cc}1& 1\\ 0& 1\\ & & 1\\ & & & \ddots \\ & & & & 1\end{array}\right),$$

i.e. $g$ is unipotent and fixes a hyperplane.

Theorem If $ℒ$ is a projective lattice of rank $r\ge 3$ then

$$\text{Aut}\left(ℒ\right)=P\Gamma L_r\left(𝔻\right)=P{GL}_r\left(𝔻\right)⋊\text{Gal}\left(\raisebox{1ex}{$𝔻$}\!\left/ \!\raisebox{-1ex}{$?$}\right.\right)$$

where $𝔻$ is a division ring and

$$ℒ\hspace{0.17em}\text{is the lattice of subspaces of}\hspace{0.17em}{𝔻}^r\text{.}$$

Notes and References

This is a typed copy of handwritten notes by Arun Ram from discussions with J. Bamberg and M. Givdici on 24-25/10/2012.

page history