It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.
I have a counterexample for non-reduced schemes, but how about the perfectness of the base field? So my questions are:
(1) Where does Serre's proof break down for non-perfect base fields?
(2) Does anyone know a counterexample over non-perfect fields?
NB: Serre does not state the hypothesis (maybe because at that time varieties were always integral over an algebraically closed field), but every author who quotes him states those hypothesis.
Best Answer
The arguments of Serre can be in fact made to work over any separably closed field. The result in the general case can then be deduced using Galois descent. Details can be found in Section 2 and the appendix of:
Olivier Wittenberg - On Albanese torsors and the elementary obstruction.
This is in particular shows the existence of the Albanese torsor and Albanese variety for any geometrically integral variety $X$ over any field $k$.
Note that when dealing with non-proper varieties, the Albanese variety is usually defined to be a semi-abelian variety, rather than just an abelian variety.