Degree of unirationality for del Pezzo surfaces over finite fields (Q259500)

In the paper under review, the author studies the degrees of unirational parametrizations of cubic surfaces defined over finite fields. Over an algebraically closed filed, for every del Pezzo surface \(S\), there exists a birational map \(\mathbb{P}^{2} \dashrightarrow S\). If the base field is not algebraically closed, some del Pezzo surface \(S\) may not be rational. However there might be a rational map \(\mathbb{P}^{2} \dashrightarrow S\). This kind of rational map is called a unirational parametrization. The first main result of this paper is that every del Pezzo surface of degree 4 over a finite field has a unirational parametrization of degree 2. This result was known by \textit{Yu. I. Manin} when the base field has more than 22 elements. [Cubic forms. Algebra, geometry, arithmetic. Translated from Russian by M. Hazewinkel. London: North- Holland Publishing Company; New York: American Elsevier Publishing Company, Inc (1974; Zbl 0277.14014)] The author removes the condition on the base field. Then it easily follows that every cubic surface, which is a del Pezzo surface of degree 3, containg a rational excetional curve over a finite field has a unimodular parametrization of degree 2. Conversely, the author also prove that a cubic surface which is eqipped with a unirational parametrization of degree 2 contains a rational exceptional curve if the characteristic of the base field is odd. Moreover for a minimal cubic surface over a finite field, there exists a unirational parametrization of degree 6.