Nondiscreteness of $F$-thresholds

For every integer g > 1 and prime p > 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 Mustata-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).