Convex bodies appearing as Okounkov bodies of divisors
Convex bodies appearing as Okounkov bodies of divisors
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凸体表现为约数的奥孔科夫体
DOI：
10.1016/j.aim.2012.01.013
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发表时间：
2010
影响因子：
1.7
通讯作者：
C. Maclean
中科院分区：
文献类型：
--
作者：
Alex Kuronya;Victor Lozovanu;C. Maclean
基于Okounkov（Okounkov，1996 [15]，2003 [16]）的工作，Lazarsfeld和Mustafeld（2009）[13]以及Kaveh和Khovanskii（预印本）[10]独立地将凸体（称为Okounkov体）与关于完整标志的光滑投影簇上的大因子相关联。在本文中，我们考虑以下问题：什么可以说的一套凸体出现奥肯科夫机构？我们首先证明了在光滑射影簇上的大线丛的奥肯科夫体关于容许旗的凸体集合是可数的。然后，我们给出了一个完整的特征的一组凸体，出现作为奥肯科夫机构的R-因子光滑的射影曲面。这样的奥肯科夫体总是多边形，满足一定的组合准则。最后，我们构造了两个非多面奥昆科夫体的例子。在第一种方法中，我们处理的是Fano簇，线丛是充足的。在第二个，我们发现一个森梦想的空间品种，使旗的小扰动下的Okounkov体保持非多面体。
Based on the work of Okounkov (Okounkov, 1996 [15], 2003 [16]), Lazarsfeld and Mustaţă (2009) [13] and Kaveh and Khovanskii (preprint) [10] have independently associated a convex body, called the Okounkov body, to a big divisor on a smooth projective variety with respect to a complete flag. In this paper we consider the following question: what can be said about the set of convex bodies that appear as Okounkov bodies? We show first that the set of convex bodies appearing as Okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable. We then give a complete characterisation of the set of convex bodies that arise as Okounkov bodies of R-divisors on smooth projective surfaces. Such Okounkov bodies are always polygons, satisfying certain combinatorial criteria. Finally, we construct two examples of non-polyhedral Okounkov bodies. In the first one, the variety we deal with is Fano and the line bundle is ample. In the second one, we find a Mori dream space variety such that under small perturbations of the flag the Okounkov body remains non-polyhedral.