Incidences and projective geometries

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 2 June 2013


An incidence geometry is a triple (P,L,I) where P and L are sets and IP×L.

IP×L pr1P pr2pr2 L

A point pP is contained in a line L if (p,)I. A set of points SP is collinear if there exists L such that if pS then (p,)I.

Often it is convenient to

identifyLwith the set of points pr1 (pr2-1()) .


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

ifp1,p2P andp1p2 then there exists a uniqueLwith (p1,)I and(p2,)I.

The line =(p1,p2) containing p1 and p2 is the line connecting p1 and p2.

A subspace is a subset SP such that

ifp1,p2S thenpr1 (pr2-1((p1,p2))) S.

A subspace is SP which contains any line connecting two of its points.


Let be a partially ordered set and let x,y. The join, or supremum, or least upper bound of x and y is

xy=sup{x,y} insuch that
(a) sup{x,y}x and sup{x,y}y, and
(b) If z and zx and zy then zsup{x,y}.

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

xy=inf{x,y}in such that
(a) inf{x,y}x and inf{x,y}y, and
(b) If z and zx and zy then zinf{x,y}.

A lattice is a partially ordered set such that

ifx,ythen xyandxyexist in .

A modular lattice is a lattice such that

ifx,zand xz thenx(yz)= (xy)z.

Projective lattices

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

An atom is a such that there does not exist a with 0<a<a.

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

A maximal chain is a maximal length sequence 0<a1<a2<<a<1 in .

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

Let be a ranked lattice and let a. The rank of a is i if there exists a maximal chain

0<a1<a2<< a<1with ai=a.Write rank(a)=?

A projective lattice is an atomic ranked modular lattice such that

ifx,ythen rank(xy)+ rank(xy)= rank(x)+ rank(y).

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

IP×L pr1P pr2pr2 L

such that

(a) If p1,p2P and p1p2 then there exists a unique line (p1,p2)L containing p1 and p2,
(b) If p1,p2,p3P are noncollinear and is a line intersecting (p1,p3) and (p2,p3) then there exists p6P contained in and (p1,p2). p1 p2 p3 p6 (p1,p2) (p1,p3) (p2,p3)
(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 be the subspace lattice of (P,L,I). Then

{projective geometries} {projective lattices} (P,L,I)

is a bijection.


An automorphism of (P,L,I) is

gSym(P)× Sym(L)such that gI=I.

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

g ( Sym(P)× Sym(L) ) Sym(I).

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

IP×L pr2P pr1pr2 L isG-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 is a projective lattice of rank r3 then

Aut()=PΓ Lr(𝔻)=P GLr(𝔻) Gal(𝔻?)

where 𝔻 is a division ring and

is the lattice of subspaces of𝔻r.

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.

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