Characterizations of Classical Algebraic Varieties in Some Papers of Giuseppe Tallini and Some Personal Memories of a Friend

Abstract

One of the fundamental results of this century on finite geometries is undoubtedly the theorem by B.Segre characterizing rational points on a conic in finite Desarguesian planes of odd order q just as a set of q + 1 and three by three noncollinear points.Apart from its intrinsic scientific value, this celebrated and surprising result played a central role in the development of the geometry over Galois fields since it was the starting point of a new, wide and still active research field: to characterize rational points of classical algebraic varieties over finite fields in terms of their combinatorial and incidence properties. Giuseppe Tallini, who was among the favourite students of B.Segre and succeeded him as full professor at the University of Rome La Sapienza, was one of the leading geometers who gave great contributions to the above mentioned research area. I will now discuss the following Tallini's papers in this context.