How to determine the sign of a 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 non-constant polynomials) is completely determined by the sign of its value on the 'last key form' (key forms being the avatar of 'key polynomials' of valuations (introduced by [Maclane, 1936]) in 'global coordinates'). The proof involves computations related to the cone of curves on certain compactifications of C^2 and gives 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 [Favre-Jonsson, 2007]. A by-product of these arguments is a characterization of valuations which 'determine' normal compactifications of C^2 with one irreducible curve at infinity in terms of an associated 'semigroup of values'.