AI 中文总结
本文给出光滑射影曲线乘积上正次外部张量积的无穷小 Newton-Okounkov 体的显式 Minkowski 分解公式，并据此得到 Minkowski 包含取等的精确判据及有限多个体在共同很一般轨迹上的实现。
AI 中文摘要
即使在具有简单乘积结构的簇上，一般无穷小 Newton-Okounkov 体的显式计算也是困难的。我们给出了任意维数下光滑射影曲线乘积上正次外部张量积的显式公式。将度向量的递减重排记为 $d^\downarrow=(d_1^\downarrow,\ldots,d_n^\downarrow)$，并设 $d_{n+1}^\downarrow=0$，则该体具有显式的 Minkowski 分解 $\sum_{j=1}^n(d_j^\downarrow-d_{j+1}^\downarrow)S_j^{(n)}$，其中 $S_j^{(n)}$ 是下文定义的嵌入单纯形。利用这一描述，我们给出了 Minkowski 包含中取等号的精确判据。一个同时重标号的论证还允许在对其度向量进行独立的递减重排后，在旗簇的公共一般很一般轨迹上实现有限多个这样的体。
英文摘要
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.
