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Given a divisorial discrete valuation 'centered at infinity' on C[x,y], weshow that its sign on C[x,y] (i.e. whether it is negative or non-positive onnon-constant polynomials) is completely determined by the sign of its value onthe 'last key form' (key forms being the avatar of 'key polynomials' ofvaluations (introduced by [Maclane, 1936]) in 'global coordinates'). The proofinvolves computations related to the cone of curves on certaincompactifications of C^2 and gives a characterization of the divisorialvaluations centered at infinity whose 'skewness' can be interpreted in terms ofthe 'slope' of an extremal ray of these cones, yielding a generalization of aresult of [Favre-Jonsson, 2007]. A by-product of these arguments is acharacterization of valuations which 'determine' normal compactifications ofC^2 with one irreducible curve at infinity in terms of an associated 'semigroupof values'.

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