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 [dFEI08]. 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.

Toric completions and bounded functions on real algebraic varieties

“…The properties of filtrations obtained in this way and further generalizations have also been studied in complex algebraic geometry (see Mondal [8]). For us, this question is particularly relevant in the context of positive polynomials and the moment problem, as it concerns possible degree cancellations in sums of positive polynomials, as explained in Section 5.…”

Algebraicity of normal analytic compactifications of ℂ2 with one irreducible curve at infinity

“…Degree-like functions and compactifications. In this subsection we recall from [Mon14] the basic facts of compactifications of affine varieties via degree-like functions. Recall that X = C 2 in our notation; however the results in this subsection remains valid if X is an arbitrary affine variety.…”

How to determine the sign of valuation on ℂ[x,y]

“…Recall that a divisorial discrete valuation (Definition 2.2) ν on R is centered at infinity iff ν(f ) < 0 for some f ∈ R, or equivalently iff there is an algebraic completion X of X := Spec R (i.e. X is a complete algebraic varieties containing X as a dense open subset) and an irreducible component C of X \ X such that ν is the order of vanishing along C. On the other hand, one way to construct algebraic completions of the affine variety X is to start with a degree-like function on R (the terminology is from [Mon10b] and [Mon10a]), i.e. a function δ : R → Z ∪ {−∞} which satisfy the following 'degree-like' properties:…”

“…A fundamental class of degree-like functions are divisorial semidegrees which are precisely the negative of divisorial discrete valuations centered at infinity -they serve as 'building blocks' of an important class of degree-like functions (see [Mon10b], [Mon10a]). Therefore, a natural question in this context is:…”