Zero cycles on homogeneous varieties

In this paper we study the group A₀(X) of zero-dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety X. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras which we construct. This allows us to show A₀(X)=0 for certain classes of homogeneous varieties for groups of each of the classical types, extending previous results of Swan/Karpenko, of Merkurjev, and of Panin.     

Remarks on varieties of involution bands

In the present paper, we study varieties consisting of bands (idempotent semigroups) endowed with an involutorial antiautomorphism * as a fundamental operation. Our principal aim is here to provide an insight to some classes of these varieties from the structural point of view, especially in terms of semilattice decompositions, subdirect products and ideal extensions. In the course of such considerations, we shall extend the result of C. L. Adair [1], who described the lattice of all varieties of bands with a regular involution (i.e. with the identity x = xx ^* x) We depict a broader lattice of involution band varieties, which incorporates Adair’s lattice.     

Algebraic cycles on Prym varieties

This article proposes a generalization of tautological rings introduced by Beauville and Moonen for Jacobians. The main result is that, under certain hypotheses, the special subvarieties of Prym varieties are algebraically equivalent and their classes belong to the tautological ring.     

Minimal varieties of semirings with involution

We give a complete list of minimal varieties (varieties having no proper subvarieties) of semirings equipped with an involutorial antiautomorphism as a unary fundamental operation. Received November 1, 1999; accepted in final form March 6, 2000.     

Zero cycles on certain surfaces in arbitrary characteristic

Letk be a field of arbitrary characteristic. LetS be a singular surface defined overk with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation is finite dimensional. We give numerical conditions under which the Chow group of zero cycles ofS is finite dimensional.     

Zero cycles on conic fibrations and a conjecture of Bloch

Let X be a smooth projective surface over a number field k. Let (CH₀(X)) denote the Chow group of zero-cyles modulo rational equivalence on X. Let CH₀(X) be the subgroup of CH ₀(X) consisting of classes which vanish when going over to an arbitrary completion of k. Bloch put forward a conjecture asserting that this group is isomorphic to the Tate-Shafarevich group of a certain Galois module atttached to X. In this paper, we disprove this general conjecture. We produce a conic bundle X over an elliptic curve, for which the group (CH₀(X) is not zero, but the Galois-theoretic Tate-Shafarevich group vanishes.     

Some elementary theorems about algebraic cycles on abelian varieties

The structure of the group of 0-cycles modulo rational equivalence on ann-dimensional abelian varietyA over an algebraically closed fieldk is studied. This group forms an augmented -algebra under Pontryagin product, with augmentation given by the degree map. The (n+1)-st power of the augmentation idealI(=0-cycles of degree 0) is shown to be zero, while for suitablek (e.g.k=complex numbers) then-th power ofI is non-zero. As corollary, every 0-cycle of degree 0 is shown to be rationally equivalent to a sum of intersections of divisors. Partial results, analogous to the isogeny betweenA and Pic⁰ A, are proved relating quotientsI ^*r /I ^*r+1 to cycles of codimensionr onA.Partially supported by the C.N.R.S. and by a Nato Fellowship