Projective completions of affine varieties via degree-like functions

TL;DR

This paper explores projective completions of affine varieties using degree-like functions, generalizing toric variety constructions and extending valuation finiteness results to broader algebraically closed fields.

Contribution

It introduces a new class of projective completions based on multiplicative filtrations, generalizes existing valuation finiteness theorems, and provides formulas for divisor pull-backs and intersection numbers.

Findings

Generalized toric variety construction from convex polytopes

Extended valuation finiteness property to arbitrary algebraically closed fields

Derived formulas for divisor at infinity and intersection matrices

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 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.