Natural dualities, nilpotence and projective planes

Affine and projective planes

Smooth Projective Planes

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to CP. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory ...

Projective Planes in Algebraically

We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. The main technique used, which is motivated by classical projective geometry, is that a particul...

Extending Partial Projective Planes

If a finite partial projective plane can be extended to a projective plane of order n, then it can be extended by a sequence of one line extensions. We describe necessary conditions for such an extension to exist, and restrictions which can be imposed on the intermediate sequence of extensions. This method can be used to attempt to construct a projective plane of non-prime-power order. 1. Parti...

Fake Projective Planes

1.1. A fake projective plane is a smooth compact complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane. Such a surface is known to be projective algebraic and it is the quotient of the (open) unit ball B in C (B is the symmetric space of PU(2, 1)) by a torsion-free cocompact discrete subgroup of PU(2, 1) whose Euler-Poincaré cha...