Homogeneous varieties - zero cycles of degree one versus rational points

Examples of projective homogeneous varieties over the field of Laurent series over p-adic fields which admit zero cycles of degree one and which do not have rational points are constructed. Let k be a field and X a quasi-projective variety over k. Let Z0(X) denote the group of zero cycles on X and deg : Z0(X) → Z the degree homomorphism which associates to a closed point x of X, the degree [k(x) : k] of its residue field. We study the question: for what classes X of varieties (respectively, what classes of fields k) is it true that for X ∈ X, if X admits a 0-cycle of degree one, then X has a rational point. If X is a curve of genus zero or one over any field k, then X has a rational point once it admits a zero cycles of degree one. However, the question is more reasonable to ask for classes of rational varieties or homogeneous varieties. In the setting of rational varieties, there are examples, due to Colliot-Thélène and Coray [CTC] of conic bundles over the projective line over a p-adic field with a zero cycle of degree one, which have no rational points. We shall list from literature some questions in this direction for homogeneous varieties. Q(HPr) (Veisfeiler)[V] Let X be a projective homogeneous variety under a connected linear algebraic group defined over a field k. If X has a zero cycle of degree one, does X have a rational point?