## Incidences and projective geometries

Last update: 2 June 2013

## Incidence

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

$I⊆P×L ⟶pr1P pr2↓pr2 L$

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

Often it is convenient to

$identify ℓ∈L with the set of points pr1 (pr2-1(ℓ)) .$

## Subspaces

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

$if p1,p2∈P and p1≠p2 then there exists a unique ℓ∈L with (p1,ℓ)∈I and (p2,ℓ)∈I.$

The line $\ell =\ell \left({p}_{1},{p}_{2}\right)$ 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

$if p1,p2∈S then pr1 (pr2-1(ℓ(p1,p2))) ⊆S.$

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

$x∨y=sup{x,y} in ℒ such that$
 (a) $\text{sup}\left\{x,y\right\}\ge x$ and $\text{sup}\left\{x,y\right\}\ge y,$ and (b) If $z\in ℒ$ and $z\ge x$ and $z\ge y$ then $z\ge \text{sup}\left\{x,y\right\}\text{.}$

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

$x∧y=inf{x,y} in ℒ such that$
 (a) $\text{inf}\left\{x,y\right\}\le x$ and $\text{inf}\left\{x,y\right\}\le y,$ and (b) If $z\in ℒ$ and $z\le x$ and $z\le y$ then $z\le \text{inf}\left\{x,y\right\}\text{.}$

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

$if x,y∈ℒ then x∨y and x∧y exist in ℒ.$

A modular lattice is a lattice $ℒ$ such that

$if x,z∈ℒ and x≤z then x∨(y∧z)= (x∨y)∧z.$

## 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

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 projective lattice is an atomic ranked modular lattice such that

$if x,y∈ℒ then rank(x∨y)+ rank(x∧y)= rank(x)+ rank(y).$

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

$I⊆P×L ⟶pr1P pr2↓pr2 L$

such that

 (a) If ${p}_{1},{p}_{2}\in P$ and ${p}_{1}\ne {p}_{2}$ then there exists a unique line $\ell \left({p}_{1},{p}_{2}\right)\in L$ containing ${p}_{1}$ and ${p}_{2},$ (b) If ${p}_{1},{p}_{2},{p}_{3}\in P$ are noncollinear and $\ell$ is a line intersecting $\ell \left({p}_{1},{p}_{3}\right)$ and $\ell \left({p}_{2},{p}_{3}\right)$ then there exists ${p}_{6}\in P$ contained in $\ell$ and $\ell \left({p}_{1},{p}_{2}\right)\text{.}$ ${p}_{1} {p}_{2} {p}_{3} {p}_{6} \ell \left({p}_{1},{p}_{2}\right) \ell \left({p}_{1},{p}_{3}\right) \ell \left({p}_{2},{p}_{3}\right) \ell$ (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 $\left(P,L,I\right)\text{.}$ Then

${projective geometries} ⟷ {projective lattices} (P,L,I) ⟼ℒ$

is a bijection.

## Automorphisms

An automorphism of $\left(P,L,I\right)$ is

$g∈Sym(P)× Sym(L) such that gI=I.$

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

$g∈ ( Sym(P)× Sym(L) ) ∩Sym(I).$

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

$I⊆P×L ⟶pr2P pr1↓pr2 L is G-equivariant.$

A homology is a matrix $g$ conjugate to

$( a 1 ⋱ 1 ) ,$

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

An elation is a matrix $g$ conjugate to

$( 11 01 1 ⋱ 1 ) ,$

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

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

$Aut(ℒ)=PΓ Lr(𝔻)=P GLr(𝔻)⋊ Gal(𝔻?)$

where $𝔻$ is a division ring and

$ℒ is the lattice of subspaces of 𝔻r.$

## 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.