Projective surfaces over a finite field

Abstract:

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 $\Omega$ 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 $\Omega$. Using this theorem we characterize ${\mathbf {P}}_k^2$ as a topological space, and we show that for any two irreducible plane curves $C$, $C’$ there is a homeomorphism from ${\mathbf {P}}_k^2$ onto itself taking $C$ onto $C’$.