Nondiscreteness of $F$-thresholds

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).

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Received 30 January 2019

Accepted 3 August 2019

Published 17 February 2021