A short proof of Castelnuovo’s criterion of rationality

Abstract

We give a new proof in positive characteristic of Castelnuovo's criterion of rationality of algebraic surfaces. We use crystalline cohomology and the de Rham-Witt complex as a substitute for the transcendeal methods of Kodaira. Let X be a nonsingular complete algebraic surface over an algebraically closed field k. Let pa(X) = X(Ox) 1 and let Pn(X) = ho(X, X where Kx is the canonical line bundle. These are birational invariants of X. THEOREM 1. X is rational (birationally isomorphic to the projective plane) if and only if Pa(X) = P2(X) = 0. Over the complex numbers, this is a classical result of Castelnuovo. A modem proof was given by Kodaira [10]. Zariski [16] gave the first proof in characteristic p > 0; other proofs were given by M. Artin (unpublished) and G. Kurke [11]. In this paper, we give a short proof of the theorem in positive characteristic. Our proof is closely related to Artin's; however, by using crystalline cohomology and the de Rham-Witt complex, we are able to give a better approximation in characteristicp to the elegant Kodaira proof in characteristic zero. Our debt to the work of Nygaard [13] and Artin and Swinnerton-Dyer [3] will be obvious. After the first version of this paper was written, I learned that S. Mori has also given a new proof of Castelnuovo's criterion, using his theory of extremal rational curves. I believe that this proof complements his, and have indicated briefly in certain places how (according to taste) one may substitute arguments from his proof for those given here. PROOF OF THEOREM 1. Serre [14] has given an exposition of Kodaira's proof which shows (in all characteristics) that if X is a nonsingular complete algebraic surface free from exceptional curves of the first kind withpa(X) = P2(X) = 0, then either X is rational, or X satisfies (1) Pic(X) is an infinite cyclic group generated by Kx, (2) I-KI consists of irreducible curves of arithmetic genus 1, and diml-KI > 1. Received by the editors December 14, 1979 and, in revised form, February 19, 1980. AMS (MOS) subject classifications (1970). Primary 14J10; Secondary 14F30.