Newton–Okounkov bodies of exceptional curve valuations

Abstract

We prove that the Newton-Okounkov body of the flag E • := {X = X r ⊃ E r ⊃ {q}}, defined by the surface X and the exceptional divisor E r given by any divisorial valuation of the complex projective plane P 2 , with respect to the pull-back of the line-bundle O P 2 (1) is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular.

Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

“…; taking into account that the above dimensions depend only on local data, the equality β g+1 = [vol(ν)] −1 follows as in [19,Remark 2.3].…”

“…In [17] we proved that the fact that ν belongs to N is equivalent to that of the cone of curves N E(X) is regular, and we gave a simple characterization of this fact. Even more, we are able to compute the Seshadri-like constant with respect to a line divisor for valuations in N [18], and to explicitly give the Newton-Okounkov bodies of flags where the valuation given by the divisor E r belongs to N [19].…”

Discrete Equivalence of Non-positive at Infinity Plane Valuations

“…For instance, the Seshadri-related constants introduced in [4] (see [7] for the plane case) can be explicitly described for NPI valuations [12,13] and are essential for providing some evidences to the valuative Nagata conjecture (which implies the Nagata classical conjecture) [15] (see also [7]). Newton-Okounkov bodies [3,19,22] of flags determined by valuations of this type can also be explicitly described [13,16]. Finally, NPI divisorial valuations characterize those rational surfaces Z (given by simple sequences of point blowing-ups) with polyhedral cone of curves and minimum number of extremal rays.…”

Global geometry of surfaces defined by non-positive and negative at infinity valuations

“…In particular, the first components of the vertices of the bodies ∆ ν Y• (D) are those values t where the ray [D t ] crosses into a different Zariski chamber [82]. Furthermore, [65] considers surfaces X defined by divisorial valuations ν of the function field of P 2 centered at some point p ∈ P 2 and flags E • = {X ⊃ E ⊃ {q}} where E is the exceptional divisor defining ν (= ν E ). In that paper, the authors show that the exceptional curve valuations centered at p and valuations ν E• defined by flags E • are the same thing.…”

“…This concept is strongly involved in a valuative conjecture formulated in [64] (see also [38]) which implies the well-known Nagata conjecture. In addition,μ L (ν E ) can be geometrically understood asμ L (ν E ) = sup{s > 0 | L * − sE is big} (when k has characteristic zero), where L * is the pull-back of L on the surface defined by ν E , which establishes a relation with the Newton-Okounkov bodies [15,28,65]. When ν E is non-positive at infinity,μ L (ν E ) have been computed explicitly in [64].…”