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

Mondal, Pinaki

Asian J. Math. 18, No. 4, 573-602 (2014).

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.

MSC:

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

Keywords:

compactification; degree-like functions; divisorial valuations; Rees valuations; normalization at infinity

Citations:

Zbl 1162.14023