Projective completions of affine varieties via degree-like functions

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 var...

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Bibliographic Details
Published in:The Asian journal of mathematics 2014, Vol.18 (4), p.573-602
Main Author: Mondal, Pinaki
Format: Article
Language:English
Subjects:
13A18
13A30
14M25
14M27
Compactification
Degree-like functions
divisorial valuations
normalization at infinity
Rees valuations
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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 “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.
ISSN:1093-6106
1945-0036
DOI:10.4310/AJM.2014.v18.n4.a1