  In this note we study linear systems on complete toric varieties $X$ with an
invariant point, whose orbit under the action of the automorphism group of $X$
contains the dense torus $T$ of $X$. We give a characterization of such
varieties in terms of its defining fan and introduce a new definition of
expected dimension of linear systems which consider the contribution given by
certain toric subvarieties. Finally, we study degenerations of linear systems
on these toric varieties induced by toric degenerations.
