The Class of Non-Desarguesian Projective Planes is Borel Complete

For every infinite graph $\Gamma$ we construct a non-Desarguesian projective plane $P^*_{\Gamma}$ of the same size as $\Gamma$ such that $Aut(\Gamma) \cong Aut(P^*_{\Gamma})$ and $\Gamma_1 \cong \Gamma_2$ iff $P^*_{\Gamma_1} \cong P^*_{\Gamma_2}$. Furthermore, restricted to structures with domain $\omega$, the map $\Gamma \mapsto P^*_{\Gamma}$ is Borel. On one side, this shows that the class of countable non-Desarguesian projective planes is Borel complete, and thus not admitting a Ulm type system of invariants. On the other side, we rediscover the main result of [15] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete

Conic Blocking Sets in Desarguesian Projective Planes

a conic blocking set to be a set of lines in a Desarguesian projective plane such that all conics meet these lines. Conic blocking sets can be used in determining if a collection of planes in projective three-space forms a flock of a quadratic cone. We discuss trivial conic blocking sets and conic blocking sets in planes of small order. We provide a construction for conic blocking sets in planes of non-prime order, and we make additional comments about the structure of these conic blocking sets in certain planes of even order

Transitive projective planes and insoluble groups

Suppose that a group G acts transitively on the points of P, a finite non-Desarguesian projective plane. We prove that if G is insoluble then G/O(G) is isomorphic to SL2(5) or SL2(5).2

Semiaffine spaces

In this paper we improve on a result of Beutelspacher, De Vito & Lo Re, who characterized in 1995 finite semiaffine spaces by means of transversals and a condition on weak parallelism. Basically, we show that one can delete that condition completely. Moreover, we extend the result to the infinite case, showing that every plane of a planar space with atleast two planes and such that all planes are semiaffine, comes from a (Desarguesian) projective plane by deleting either a line and all of its points, a line and all but one of its points, a point, or nothing

Minimal symmetric differences of lines in projective planes

Let q be an odd prime power and let f(r) be the minimum size of the symmetric difference of r lines in the Desarguesian projective plane PG(2,q). We prove some results about the function f(r), in particular showing that there exists a constant C>0 such that f(r)=O(q) for Cq^{3/2}<r<q^2 - Cq^{3/2}.Comment: 16 pages + 2 pages of tables. This is a slightly revised version of the previous one (Thm 6 has been improved, and a few points explained

On the linearity of higher-dimensional blocking sets

A small minimal k-blocking set B in PG(n,q), q = p(t), p prime, is a set of less than 3(q(k)+1)/2 points in PC(n,q), such that every (n - k)-dimensional space contains at least one point of B and such that no proper subset of B satisfies this property. The linearity conjecture states that all small minimal k-blocking sets in PG(n,q) are linear over a subfield F(pe) of F(q). Apart from a few cases, this conjecture is still open. In this paper, we show that to prove the linearity conjecture for k-blocking sets in PG(n,p(t)), with exponent e and p(e) >= 7, it is sufficient to prove it for one value of n that is at least 2k. Further more, we show that the linearity of small minimal blocking sets in PG(2,q) implies the linearity of small minimal k-blocking sets in PG(n,p(t)), with exponent epsilon, with p(e) >= t/epsilon + 11

Projective Planes with a Doubly Transitive Projective Subplanes

none1Projective planes P of order up to q^3 with a collineation group G acting 2-transitively on a subplane of order $q$ are investigated.A. MONTINAROMontinaro, Alessandr