  We call a sheaf on an algebraic variety immaculate if it lacks any cohomology
including the zero-th one, that is, if the derived version of the global
section functor vanishes. Such sheaves are the basic tools when building
exceptional sequences, investigating the diagonal property, or the toric
Frobenius morphism.
  In the present paper we focus on line bundles on toric varieties. First, we
present a possibility of understanding their cohomology in terms of their
(generalized) momentum polytopes. Then we present a method to exhibit the
entire locus of immaculate divisors within the class group. This will be
applied to the cases of smooth toric varieties of Picard rank two and three and
to those being given by splitting fans.
  The locus of immaculate line bundles contains several linear strata of
varying dimensions. We introduce a notion of relative immaculacy with respect
to certain contraction morphisms. This notion will be stronger than plain
immaculacy and provides an explanation of some of these linear strata.
