Projective completions of affine varieties via degree-like functions

Abstract

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 “Divisorial valuations via arcs” [T. de Fernex, L. Ein, and S. Ishii, Publ. Res. Inst. Math. Sci., 44:2 (2008), pp. 425–448]. 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.