On Minkowski decomposition of Okounkov bodies on a Del Pezzo surface
Abstract
We show that on a blow up of P in 3 general points there exists a finite set of nef divisors P1 , . . . , Ps such that the Okounkov body ∆(D) of an arbitrary effective R–divisor D on X is the Minkowski sum X s ∆(D) = ai ∆(Pi ) (1) i=1 with non-negative coefficients ai ∈ R⩾0 .
References
Th. Bauer, A simple proof for the existence of Zariski decompositions on surfaces, J. Algebraic Geom. 18 (2009), 789–793.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Th. Bauer, A. Küronya, T. Szemberg, Zariski chambers, volumes, and stable base loci, J. Reine Angew. Math. 576 (2004), 209–233.
##plugins.generic.googleScholarLinks.settings.viewInGS##
H. Clemens, J. Kollár, S. Mori, Higher-dimensional complex geometry, Astérisque 166 (1989), 144 pp.
##plugins.generic.googleScholarLinks.settings.viewInGS##
M. Demazure, Surfaces de del Pezzo,I,II,III,IV,V. Singularites des surfaces, Lecture Notes in Math. 777 (1980), 21–69.
##plugins.generic.googleScholarLinks.settings.viewInGS##
K. Kaveh, A. Khovanskii, Convex bodies and algebraic equations on affine varieties, preprint, arXiv:0804.4095.
##plugins.generic.googleScholarLinks.settings.viewInGS##
A. Küronya, V. Lozovanu, C. Maclean, Convex bodies appearing as Okounkov bodies of divisors, arXiv:1008.4431v1.
##plugins.generic.googleScholarLinks.settings.viewInGS##
R. Lazarsfeld, Positivity in Algebraic Geometry I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete, 48, 49, Springer–Verlag, Berlin, 2004.
##plugins.generic.googleScholarLinks.settings.viewInGS##
R. Lazarsfeld, M. Mustata, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. 42 (2009), 783–835.
##plugins.generic.googleScholarLinks.settings.viewInGS##
P. Łuszcz-Świdecka, T. Szemberg, Minkowski decomposition of Okounkov bodies on Del Pezzo surfaces, in preparation.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Y. Tschinkel, Algebraic varieties with many rational points, Arithmetic geometry, 243–334, Clay Math. Proc., 8, Amer. Math. Soc., Providence, RI, 2009.
##plugins.generic.googleScholarLinks.settings.viewInGS##
O. Zariski, The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface, Ann. Math. 76 (1962), 560–616.
##plugins.generic.googleScholarLinks.settings.viewInGS##
