Projective completions of affine varieties via degree-like functions. (English) Zbl 1326.14126

Summary: 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 [T. de Fernex et al., Publ. Res. Inst. Math. Sci. 44, No. 2, 425–448 (2008; Zbl 1162.14023)]. 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.


14M27 Compactifications; symmetric and spherical varieties
13A18 Valuations and their generalizations for commutative rings
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics


Zbl 1162.14023
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