Minkowski decomposition and generators of the moving cone for toric varieties

Abstract

We prove that for smooth projective toric varieties, the Okounkov body of a $T$-invariant pseudo-effective divisor with respect to a $T$-invariant flag decomposes as a finite Minkowski sum of indecomposable polytopes. We prove that these indecomposable polytopes form a Minkowski base and that they correspond to the rays in the secondary fan. Moreover, we present an algorithm to find the Minkowski base.

Version 2: one more section on how to perform a Minkowski decomposition for toric polytopes Version 3: minor changes

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• Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone
• On the additivity of Newton-Okounkov bodies