Doug Hanes,
The a-Invariant of a Graded K-Algebra
We will develop the basic properties of the a-invariant of a
finitely generated graded algebra over a field, and
show how its value can be used in order to characterize the ring or its
associated projective variety. The underlying theme (I hope) will be that
what usually matters is whether the a-invariant is less than 0. In fact,
if you're interested in (non-embedded) projective varieties, this is the
only relevant question.