The arithmetic of zero cycles on surfaces with geometric genus and irregularity zero

Abstract

Let be a smooth, projective, geometrically irreducible surface over a perfect field F. Throughout this paper, it will be assumed that the geometric genus pg and the irregularity q of both vanish. Denote the separable closure of F by F. Let X=X| be the surface obtained from by base extension. It will also be assumed that the group Ao(X) of rational equivalence classes of zero cycles of degree zero on X vanishes. This is a technical hypothesis which could presumably be eliminated at the expense of working with Ker(Ao(X)~Ao(X)). For want of a better name, and for ease in stating various results, any surface which satisfies these three hypotheses will be called a pseudo-rational surface. Bloch [2] has conjectured that the vanishing of Ao(X) should follow from the assumption that Po = q = 0. This was proven by Bloch, Kas, and Lieberman [5] for all such surfaces which are not of general type; i.e., which have Kodaira dimension less than 2. It has also been proven for particular surfaces of general type by Inose and Mizukami [23], Barlow [1], and Keum [25]. Consequently, the class of pseudo-rational surfaces includes: rational surfaces, Enriques surfaces, elliptic surfaces with q=0, the classical Godeaux surface, Burniat-Inoue surfaces, Campedelti surfaces, and the surfaces of Barlow and of Keum. This paper will study Ao(X) for pseudo-rational surfaces defined over fields of number theoretic interest. Bloch [4] introduced K-theoretic techniques into the study of zero cycles on rational surfaces. His work was extended [7, 12, 15, 27, 28, 32, 36] to achieve a thorough understanding of such cycles. Colliot-Th616ne and Raskind [10] developed this machinery further to study codimension two cycles on any variety. The author [14] applied these techniques to Enriques surfaces. Raskind [30] used them to study zero cycles on pseudo-rational surfaces. One of the main results of this paper is a new proof of the following theorem of Raskind [31].