Minkowski Decompositions for Generic Infinitesimal Newton-Okounkov Bodies of Arbitrary-Degree External Tensor Products on Products of Curves

Abstract

Explicit computations of generic infinitesimal Newton--Okounkov bodies are difficult even for varieties with simple product structure. We give an explicit formula in arbitrary dimension for positive-degree external tensor products on products of smooth projective curves. Writing $d^\downarrow=(d_1^\downarrow,\ldots,d_n^\downarrow)$ for the decreasing rearrangement of the degree vector and setting $d_{n+1}^\downarrow=0$, the body admits the explicit Minkowski decomposition $\sum_{j=1}^n(d_j^\downarrow-d_{j+1}^\downarrow)S_j^{(n)}$, where the $S_j^{(n)}$ are the embedded simplices defined below. Using this description, we give a sharp criterion for equality in the Minkowski inclusion. A simultaneous relabeling argument also allows finitely many such bodies to be realized on a common very general locus of flags after independent decreasing rearrangements of their degree vectors.