NEWTON-OKOUNKOV BODIES OF EXCEPTIONAL CURVE PLANE VALUATIONS NON-POSITIVE AT INFINITY

In this note we announce a result determining the Newton–Okounkov bodies of the line bundle \(\mathcal {O}_{\mathbb {P}^2} (1)\) with respect to exceptional curve plane valuations non-positive at infinity.


Introduction
Newton-Okounkov bodies were introduced by Okounkov [19,20,21] and independently developed in greater generality by Lazarsfeld and Mustaµ [18], on the one hand, and Kaveh and Khovanskii [13], on the other.
The key idea is to associate a convex body to a big divisor on a smooth irreducible normal projective variety X, with respect to a specic ag of subvarieties of X, via the corresponding valuation on the function eld of X.This turns out to be a good way to relate the convex geometry of that object with positivity aspects on the side of the algebraic geometry.More specically, Newton-Okounkov bodies seem to be suitable to explain, from their convex structure, the asymptotic behavior of the linear systems given by the divisor and the valuation, as well as the structure of the Mori cone of X and positivity properties of divisors on X [2,14,15,16,17].
The computation of Newton-Okounkov bodies is a very hard task and, sometimes, their behaviour is unexpected, see Küronya and Lozovanu [17].The case when the underlying variety is a surface is also very hard but there exist some known results which can help.We know that they are polygons with rational slopes and can be computed from Zariski decompositions of divisors.Very recently, Ciliberto et al. studied in [4] the Newton-Okounkov bodies with respect to exceptional curve plane valuations ν, dened by divisorial valuations ν with only one Puiseux exponent, centered at a point p in P 2 := P 2 C , where C stands for the complex numbers.These valuations have both rank and rational rank equal to 2 and their transcendence degree equals zero.It is proved in [4] that the Newton-Okounkov bodies of the line bundle O P 2 (1), with respect to exceptional curve plane valuations, are triangles or quadrilaterals, where the vertices are given by the dening Puiseux exponent β , an asymptotic multiplicity μ corresponding with ν and a value in the segment [0, μ/β ].The asymptotic multiplicity μ can also be used to formulate a generalization of Nagata's conjecture [6], see also [11].The exact value of μ is only known in some cases, including when β < 7 + 1/9.
In this note we only announce a result which determines the Newton-Okounkov bodies of the previously mentioned line bundle with respect to exceptional curve 2010 Mathematics Subject Classication.Primary: 14C20.Key words and phrases.Newton-Okounkov bodies; Flags; Exceptional curve plane valuations; Plane valuations non-positive at innity .The rst three authors were partially supported by the Spanish Government Ministerio de Economía y Competitividad (MINECO), grants MTM2012-36917-C03-03 and MTM2015-65764-C3-2-P, as well as by Universitat Jaume I, grant P1-1B2015-02.plane valuations non-positive at innitythe proof and additional details will be published in a forthcoming paper.
Exceptional curve plane valuations non-positive at innity are a large class of exceptional curve plane valuations, that can have any number of Puiseux exponents and are dened by ags X ⊃ E ⊃ {q}, where E is the last exceptional divisor obtained after a simple nite sequence of point blowing-ups starting at P 2 , and denes a plane divisorial valuation ν E which is non-positive at innity, cf.Galindo and Monserrat [10].The valuations ν E are centered at innity (see Favre and Jonsson [9]) and present a behavior close to that of plane curves with only one place at innity (see Abhyankar and Moh [1], and Campillo, Piltant and Reguera [3]).
We nish this introduction being more specic and saying that all the mentioned Newton-Okounkov bodies are triangles and we will give their vertices explicitly.Moreover, the anticanonical Iitaka dimension of innitely many of the considered surfaces X is −∞ and, in addition, their Picard numbers are arbitrarily large.
Recall that the number of vertices of the Newton-Okounkov body dened by a ag and a big divisor on a surface X is bounded by 2ρ + 2, ρ being the Picard number of X [17], but the above mentioned results in [4] suggest that the bound could be applied even if we consider the ag on a projective model dominating X (and the Newton-Okounkov body associated to the pull-back of a big divisor on X).Our result can be regarded as new evidence supporting this conjecture.
The results presented in this short note were obtained during a visit of the fourth author to the University Jaume I. Previous studies and a large number of computations were done during the workshop Positivity and valuations held on February 2016 at the CRM in Barcelona.The authors wish to thank J. Roé and A. Küronya for stimulating their interest in Newton-Okounkov bodies as well as for their helpful comments and for pointing out a more customary name for our valuations.

The general setting
Let X be a smooth projective variety of dimension n over C. We will write K(X) for the function eld of X.Let us x a ag of subvarieties such that each Y i ⊂ X is irreducible, of codimension i and smooth at q.The point q ∈ X is called the center of the ag.
One may associate to the ag Y • a discrete valuation of rank n as follows.First, let g i = 0 be the equation of Y i in Y i−1 in a Zariski open set containg q, which is possible since Y i has codimension i.Then, for f ∈ K(X) we dene , where Then the map ν Y• : K(X) \ {0} → Z n lex dened by the sequence of maps ν i , 1 ≤ i ≤ n, as ν Y• := (ν 1 , . . ., ν n ) is a rank n discrete valuation and any maximal rank valuation comes from a ag [4, Th. 2.9].Given a ag Y • and a Cartier divisor D on X, the following subset of R n + : where { • } stands for the closed convex hull, is called to be the Newton-Okounkov body of D with respect to Y • .
Newton-Okounkov bodies are convex bodies such that where vol R n means Euclidean volume and Moreover, given D = D two big divisors on X, they are numerically equivalent if and only if the associated Newton-Okounkov bodies coincide for all admissible ags on X [12].Furthermore, in the case of surfaces, D and D are numerically equivalent (up to negative components in the Zariski decomposition that do not go through q) if and only if the associated Newton-Okounkov bodies coincide for all ags centered at q, cf.Roé [22].

Exceptional curve plane valuations and Newton-Okounkov bodies
In this section, we introduce the family of ags for which we are interested in computing Newton-Okounkov bodies.Let P 2 be the complex projective plane, and p any point in P 2 .Let R be the local ring of P 2 at p, and write F for the eld of fractions of R. Valuations ν of F centered at R are in one-to-one correspondence with simple sequences of point blowing-ups whose rst center is p [23, p. 121].
The cluster of centers of η will be denoted by C = {p = p 1 , p 2 , . ..} and we say that a point p i is proximate to p j , i > j, written p i → p j , whenever p i belongs to the strict transform of the exceptional divisor E j obtained by blowing-up p j .These valuations were classied by Spivakovsky in [23].We are interested in the class of exceptional curve valuations (in the terminology of Favre and Jonsson [8]) which corresponds to Case 3 in [23] and to type C of Delgado, Galindo and Núñez [5].These valuations are characterized by the fact that there exists a point p r ∈ C such that p i → p r for all i > r.
Notice that if we consider the surface X r obtained after blowing-up p r and the ag , where π : X r → X 0 is the composition of the rst r point blowing-ups in η, z r = 0 a local equation for E r and π * (f )/z ν 1 (f ) r may be seen as a function on E r .Notice that where (• , •) q denotes the intersection multiplicity at q.
The divisor E r is dened by a map π : X r → P 2 .The intersections of the strict transforms of the exceptional divisors in X r are represented by the so-called dual graph of π (or of ν Er ).The geodesic of the dual graph is dened to be the set of edges (and vertices) in the path joining the vertices corresponding to E 1 and E r .
Additionally, for i = 1, . . ., r, ϕ i will denote an analytically irreducible germ of curve at p whose strict transform is transversal to E i at a nonsingular point of the exceptional locus.
In spite of their importance, very few explicit examples of Newton-Okounkov bodies can be found in the literature.We are interested in an explicit computation of the Newton-Okounkov bodies of ags E • dened by exceptional curve plane valuations ν with respect to the divisor class H given by the pull-back of the linebundle O P 2 (1), which we will denote by ∆ ν (H).These Newton-Okounkov bodies were studied in [4] for valuations with only one Puiseux exponent [5].We devote the next section to announce a result which provides an explicit computation of bodies ∆ ν (H) for a large class of valuations ν as above which can have an arbitrary number of Puiseux exponents.Its proof and further details will appear elsewhere.

The result
For a start and without loss of generality, we set (X : Y : Z) projective coordinates in P 2 , L the line Z = 0, which we call the line at innity, and assume that p is the point with projective coordinates (1 : 0 : 0).Consider also coordinates x = X/Z and y = Y /Z in the ane chart dened by Z = 0 and local coordinates u = Y /X and v = Z/X around p. With the previous notation, set ν r a divisorial valuation of the fraction eld K = K(P 2 ), given by a nite sequence of point blowing-ups π : X r → X 0 = P 2 , whose rst blowing-up is at p and is dened by the exceptional divisor E r .We say that ν r is non-positive at innity whenever r ≥ 2, L passes through p 1 = p and p 2 and ν r (f ) ≤ 0 for all f ∈ C[x, y] \ {0}.Notice that our valuations are valuations centered at innity [9].Denition 4.1.An exceptional curve plane valuation ν of K centered at R is said to be non-positive at innity whenever it is given by a ag E • := {X = X r ⊃ E r ⊃ {q := p r+1 }} such that ν = ν Er is non-positive at innity.