The arithmetic of zero cycles on surfaces with geometric genus and irregularity zero
Let \(X\) be a smooth, projective, geometrically irreducible surface over a perfect field \(F\). Throughout this paper, it will be assumed that the geometric genus \(p_ g\) and the irregularity \(q\) of \(X\) both vanish. Denote the separable closure of \(F\) by \(\overline{F}\). Let \(\overline{X} = X \otimes_ F\overline{F}\) be the surface obtained from \(X\) by base extension. It will also be assumed that the group \(A_ 0(\overline X)\) of rational equivalence classes of zero cycles of degree zero on \(\overline{X}\) vanishes. This is a technical hypothesis which could presumably be eliminated at the expense of working with \(\text{Ker}(A_ 0(X)\to A_ 0(\overline{X}))\). 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. -- \textit{S. Bloch} [Commun. Algebra 3, 405-428 (1975; Zbl 0327.14002)] has conjectured that the vanishing of \(A_ 0(\overline{X})\) should follow from the assumption that \(p_ g = q = 0\). This was proven by \textit{S. Bloch, A. Kas} and \textit{D. Lieberman} [Compos. Math. 3, 135-145 (1976; Zbl 0337.14006)] for all such surfaces which are not of general type. The class of pseudo-rational surfaces includes: rational surfaces, Enriques surfaces, elliptic surfaces with \(q=0\), the classical Godeaux surface, Burniat-Inoue surfaces, Campedelli surfaces, and the surfaces of Barlow and of Keum. This paper will study \(A_ 0(X)\) for pseudo-rational surfaces defined over fields of number theoretic interest. One of the main results of this paper is a new proof of the following theorem of \textit{W. Raskind} [in Algebraic \(K\)-theory: Connections with geometry and topology, Proc. Meet., Lake Louise/Can. 1987, NATO ASI Ser., Ser. C 279, 343-388 (1989; Zbl 0709.14005)]. Theorem 0.1. Let \(X\) be a pseudo-rational surface over a non-archimedean local field \(L\) with residue field \(F\). Assume that \(X\) has very good reduction to a smooth projective surface \(Y\) over \(F\). Assume also that the characteristic of \(F\) is relatively prime to the order of the torsion subgroup of NS\((\overline{X})\). Then the specialization map on zero cycles defines a natural isomorphism \(A_ 0(X) \approx A_ 0(Y)\). [See section 3 for a precise definition of ``very good reduction.'']-- The proof of theorem 0.1 given here actually produces a somewhat stronger local result. This stronger result has the advantage that it can be used to obtain global results without any restrictive hypotheses: Theorem 0.2. Let \(X\) be a pseudo-rational surface over a number field. Then \(A_ 0(X)\) is a finite group.
rational equivalence classes of zero cycles
scientific article; zbMATH DE number 59964
scientific article