Newton–Okounkov Bodies of Exceptional Curve Plane Valuations Non-positive at Infinity

Newton–Okounkov bodies of exceptional curve valuations

“…We have deduced it from Theorem 3.12. However, in [15] we deduced the same result computing directly the Zariski decompositions of the R-divisors D t := H − tE r for all t ∈ [0, ν r (v)] and applying then Theorem 6.4 of [22]. For the sake of completeness we show now the decompositions D t = P t + N t , where P t is the positive part and N t the negative part.…”

“…This is a consequence of the fact that a ray of the form D − tC, where D is a big divisor and C a curve on X, can only cross ρ + 1 Zariski chambers. It is conjectured in [20] (see also [15]) that the bound could be applied even if the flag is considered on a projective model dominating X (and the Newton-Okounkov body associated to the pull-back of a big divisor on X). Our results can be regarded as new evidence supporting the conjecture.…”