Geometric equivalence between the Veblen and Desargues theorems and between the Pappus-Pascal and the "Three stars theorems"

Abstract

Let P(r, k) and A(2, k) be the projective r-dimensional space over the field k and the projective plane over the same field k, respectively.  Let PG(3, q) be the three-dimensional projective space over the Galois field GF(q) and AG(2, q) be the affine plane over GF(q).  Referring to the representation of P(r, k) over A(2, k) called also "Crashing" (see [1]), we prove the equivalence, from the geometric point of view, between the Veblen axiom in PG(3, q) and the Desargues theorem in AG(2, q).  Moreover, we get a representation in PG(3, q) of the Pappus-Pascal theorem in AG(2, q), consisting of a suitable con¯guration of planes, called the "Three stars theorem", which turns out to be a geometric equivalence between those two theorems. For the notations and theorems about the representation of P(r, k) over A(2, k) (and therefore in particular of PG(3, q) over AG(2, q)), we refer to the paper [1], cited in the bibliography, which the reader must know before reading this text.