Projective surfaces over a finite field

Abstract

ABsTRAcr. Let k be the algebraic closure of a finite field, and let X be an irreducible projective surface over k. Let C be a curve on X, and let Q be a finite set of closed points of X meeting each irreducible component of X. We prove that there is an irreducible curve on X whose set-theoretic intersection with C is U. Using this theorem we characterize Pk2 as a topological space, and we show that for any two irreducible plane curves C, C' there is a homeomorphism from Pk2 onto itself taking C onto C'.