  We study projective completions of affine algebraic varieties induced by
filtrations on their coordinate rings. In particular, we study the effect of
the 'multiplicative' property of filtrations on the corresponding completions
and introduce a class of projective completions (of arbitrary affine varieties)
which generalizes the construction of toric varieties from convex rational
polytopes. As an application we recover (and generalize to varieties over
algebraically closed fields of arbitrary characteristics) a 'finiteness'
property of divisorial valuations over complex affine varieties proved in the
article "Divisorial valuations via arcs" by de Fernex, Ein and Ishii (Publ.
Res. Inst. Math. Sci., 2008). We also find a formula for the pull-back of the
'divisor at infinity' and apply it to compute the matrix of intersection
numbers of the curves at infinity on a class of compactifications of certain
affine surfaces.
