Projective plane (${\mathbb{P}}^2$)

The projective plane is the two-dimensional projective space ${\mathbb{P}}^2$.

A point $[a:b:c]$ of the projective plane corresponds to the dual line $aX+bY+cZ=0$. A line given by those coefficients corresponds back to the point $[a:b:c]$. Incidence is reversed: $p\in \ell$ if and only if ${\ell }^{\ast }\in p^{\ast }$. Thus collinear point triples become concurrent line triples, preserving the essential combinatorics of incidence geometry.

Dualize a noncollinear finite point set in the real projective plane to projective lines. Their intersection vertices and consecutive line segments form an embedded graph. A vertex incident to $k$ lines has graph degree $2k$ and corresponds to a primal line containing $k$ points. Ordinary lines correspond to degree-four vertices. The faces are polygons with at least three sides.

A good dual-arrangement edge has degree-six endpoints and two triangular adjacent faces. Other edges are bad. The Euler defect identity for a projective line arrangement bounds the number of bad edges by $16t_2$: charge them to degree-four or higher-degree vertices and to nontriangular faces. This turns a count of ordinary lines into quantitative control of local triangular geometry.

A safe dual-arrangement edge is one for which every edge on a path of length at most two from either endpoint is a good dual-arrangement edge. This stronger local condition supplies a two-cell-thick triangular strip. Only a bounded multiple of the original bad edges can be unsafe: trace a shortest path to the first bad edge and reverse it through degree-six intermediate vertices.

In an embedded arrangement graph, an edge whose fixed-radius neighbourhood meets a bad edge can be charged to the first such edge along a shortest path. All intervening edges are good dual-arrangement edges, so their endpoints have bounded degree. The number of possible reversed paths is bounded in terms of the fixed radius alone. Therefore thickening a defect set by a fixed radius preserves an $O(t_2)$ edge count.

Let $t_k$ count vertices where $k$ lines meet and $f_j$ count faces with $j$ sides in a nonpencil projective line arrangement. Using Euler characteristic one, $v=\sum t_k$, $e=\sum k{t}_k$, and $2e=\sum j{f}_j$ gives $t_2=3+{\sum }_{k\ge 4}(k-3)t_k+{\sum }_{j\ge 4}(j-3)f_j$. Thus few double vertices control both high-multiplicity vertices and nontriangular faces. Discarding the face term gives the usual arrangement form of the ordinary-line inequality.

The Fano plane is the projective plane over the finite field ${\mathbb{F}}_2$. It has seven points and seven lines, each line contains three points, and two distinct lines meet in one point. A line-point incidence matrix $N$ therefore satisfies $N{N}^{\mathsf{T}}=2I+J$, making it invertible over the real numbers and useful for the transplantation theorem.