Iterative dissection of Okounkov bodies of graded linear series on \mathbbCP²

2014/09/05 by Thomas Eckl, Eckl, Thomas
Mathematics · #14C20 #14J26 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Holomorphic and Operator Theory #advanced mathematical theories #math.AG #msc:14C20 #msc:14J26

paper · pdf · doi:10.48550/arxiv.1409.1736

12 pages, 1 figure

openalex publication_date 2014/09/05 · arxiv created 2015/02/23 · arxiv updated 2015/02/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let π: X → ℙ² be the blow-up of \mathbbCP² in n points x_i in very general position, and let E_i be the exceptional divisor over x_i. For 0 ≤ n ≤ 9 we calculate Okounkov bodies of graded linear series given by sections of multiples of line bundles π^∗ O_ℙ²(d) ⊗ O_X(-m∑_i=1ⁿ E_i) with respect to a flag consisting of a line on \mathbbCP² and a point on the line in general position. Furthermore, we show what Nagata's Conjecture predicts on these Okounkov bodies when n > 9.

Citations

• Functions on Okounkov bodies coming from geometric valuations (with an appendix by Sébastien Boucksom)
• Seshadri constants via Okounkov functions and the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture

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• Newton-Okounkov bodies of exceptional curve valuations
• On Newton-Okounkov bodies of Graded Linear Series
• Minkowski decomposition of Okounkov bodies on surfaces
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