Abstract:Assume that the valuation semigroup $\Gamma (\lambda )$ of an arbitrary partial flag variety corresponding to the line bundle $\mathcal {L_\lambda }$ constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton–Okounkov body — which happens to be a rational, convex polytope — contains exactly one lattice point in its interior if and only if $\mathcal {L}_\lambda$ is the anticanonical line bundle. Furthermore, we use this unique lattice point to construct the dual polytope of the Newton–Okounkov body and prove that this dual is a lattice polytope using a result by Hibi. This leads to an unexpected, necessary and sufficient condition for the Newton–Okounkov body to be reflexive.

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中文翻译：

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部分旗形变体的牛顿-奥昆科夫体的自反性

摘要：假设通过全秩估值构造的与线丛$\mathcal {L_\lambda }$相对应的任意偏标志变体的估值半群$\Gamma (\lambda )$是有限生成和饱和的。我们使用 Ehrhart 理论来证明相关的 Newton-Okounkov 体——恰好是一个有理凸多胞体——在其内部恰好包含一个格点当且仅当 $\mathcal {L}_\lambda$ 是反规范线捆。此外，我们利用这个独特的格点构造了牛顿-奥昆科夫体的对偶多胞体，并利用 Hibi 的结果证明了这个对偶是一个格多胞体。这导致了牛顿-奥昆科夫体具有自反性的一个意想不到的充分必要条件。

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