  In an ongoing project to classify all hereditary abelian categories, we
provide a classification of Ext-finite directed hereditary abelian categories
satisfying Serre duality up to derived equivalence. In order to prove the
classification, we will study the shapes of the Auslander-Reiten components
extensively and use appropriate generalisations of tilting objects and
coordinates, namely partial tilting sets and probing of objects by
quasi-simples.
