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

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 $d^{\downarrow }=(d_1^{\downarrow },\dots ,d_n^{\downarrow })$ 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.