Arithmetic Surjectivity for Zero-Cycles

Let \(f:X\to Y\) 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 \(X_P\) over any point \(P\in Y(k_v)\) 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.