Let f:X→Y be a proper, dominant morphism of smooth varieties over a
number field k. When is it true that for almost all places v of k, the
fibre XP over any point P∈Y(kv) contains a zero-cycle of degree 1?
We develop a necessary and sufficient condition to answer this question.
The proof extends logarithmic geometry tools that have recently been
developed by Denef and Loughran-Skorobogatov-Smeets to deal with analogous
Ax-Kochen type statements for rational points.Comment: 25 pages with referee suggestions, to appear in MR