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

Abstract

Given a divisorial discrete valuation centered at infinity on C[x, y], we show that its sign on C[x, y] (i.e. whether it is negative or non-positive on C[x, y] C) is completely determined by the sign of its value on the last key form (key forms being the avatar of key polynomials of valuations [Mac36] in ‘global coordinates’). We also describe the cone of curves and the nef cone of certain compactifications of C associated to a given valuation centered at infinity, and give a characterization of the divisorial valuations centered at infinity whose skewness can be interpreted in terms of the slope of an extremal ray of these cones, yielding a generalization of a result of [FJ07]. A by-product of these arguments is a characterization of valuations which ‘determine’ normal compactifications of C with one irreducible curve at infinity in terms of an associated ‘semigroup of values’.