On the number of Frobenius-trivial vector bundles on specific curves
Let \(C\) be a smooth, connected and projective curve over an algebraically closed field \(k\) of positive characteristic. Using Cartier's theorem [\textit{N. M. Kartz}, Publ. Math., Inst. Hautes Étud. Sci. 39, 175--232 (1970; Zbl 0221.14007)] the author gets an equivalence of abelian categories between the category of Frobenius-trivial vector bundles over \(C\) and the category of representations of a certain finite generated associative algebra. In particular stable (polystable) vector bundles correspond to simple (semi-simple) representations (Proposition 3). Finally, using Letzter's algorithm which permits to determine the number of semi-simple representations of a finite generated algebra, the author provides examples, in characteristic \(2\), where infinitely many isomorphism classes of Frobenius-trivial stable vector bundles exist. This provides a counterexample to \textit{M. V. Nori}'s conjecture [Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)]. Finally the author provides examples in characteristic \(3\) showing that the aforementioned sets are finite.
scientific article; zbMATH DE number 6109278
scientific article