For every integer $g \gt 1$ and prime $p \gt 0$, we give an example of a standard graded domain $R$ (where Proj $R$ is a nonsingular projective curve of genus $g$ over an algebraically closed field of characteristic $p$), such that the set of $F$-thresholds of the irrelevant maximal ideal of $R$ is not discrete. This answers a question posed by Mustaţӑ–Takagi–Watanabe ([MTW], 2005). These examples are based on a certain Frobenius semistability property of a family of vector bundles on $X$, which was constructed by D. Gieseker using a specific “Galois” representation (analogous to Schottky uniformization for a genus $g$ Riemann surface).

中文翻译：

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$ F $阈值的非离散性

对于每个整数$ g \ gt 1 $和质数$ p \ gt 0 $，我们给出一个标准渐变域$ R $的示例（其中Proj $ R $是代数闭合的$ g $属的非奇异投影曲线）特征值$ p $）的字段，使得不相关的$ R $的最大理想值的$ F $-阈值的集合不是离散的。这回答了穆斯塔（Musta）–高木（Takagi）–渡边（Watanabe）提出的问题（[MTW]，2005年）。这些示例基于$ X $上向量束族的某些Frobenius半稳定性属性，该属性由D. Gieseker使用特定的“ Galois”表示法（类似于$ g $ Riemann曲面的肖特基均匀化）构造而成。