On Castelnuovo's criterion of rationality $p_{a} =P_{2} =0$ of an algebraic surface
Oscar Zariski · Illinois Journal of Mathematics · 1958
IntroductionLet F be a nonsingular (irreducible) algebraic surface over an algebraically closed ground field/.A theorem of Castelnuovo asserts that if the arithmetic genus pa and the bigenus P2 of F are both zero then F is a rational surface.This theorem has now been proved for fields ]c of arbitrary characteristic p, except in the case (K2)1, where K is a canonical divisor on F. In our cited paper MM (see footnote 2) we have stated that we have also a proof for the case (K2) 1, and in the present paper we shall give this proof.An immediate consequence of Castelnuovo's criterion of rationality is the well-known theorem of Castelnuovo on the rationality of plane involution.This theorem, in the case of arbitrary characteristic, is to be stated as follows"Le It(x, y) be a purely transcendenlal exlension of an algebraically closed field tc, of ranscendence degree 2, and let be a field between ] and tc(x, y), also of transcendence degree 2 over tc.If It(x, y) is a separable extension oj is a pure lranscendental extension of We shall show by an example that the condition of separability of k(x, y)/2 is essential. 2We shall make use of results established in MM for the case of surfaces F for which Pa P 0 and (K2) > 0. If (K) 1, then the Riemann- Roch inequality shows that the dimension of the anticanonical system Ka J(= -K J) is => 1.If JKa is reducible, then F is rational, by Propo- sition 7.3 of MM.We shall therefore assume that K, is irreducible.In that case we have dim K 1 (MM, Lemma 10.1), i.e., g is a pencil; it has a single base point 0, every member Ka of K has a simple point at 0, and