On the number of Frobenius-trivial vector bundles on specific curves
type: Item
instance of: scholarly article
title: On the number of Frobenius-trivial vector bundles on specific curves
zbMATH Open document ID: 1261.14019
author: João Pedro dos Santos
published in: Archiv der Mathematik
publication date: 2012-11-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 16Z05
Mathematics Subject Classification ID: 16G99
Mathematics Subject Classification ID: 18E99
zbMATH DE Number: 6109278
zbMATH Keywords: Frobenius trivial vector bundles
zbMATH Keywords: positive characteristic
zbMATH Keywords: local fundamental group scheme
zbMATH Keywords: computations in algebraic geometry
zbMATH Keywords: Letzter's algorithm
describes a project that uses: Macaulay2
MaRDI profile type: Publication
full work available at URL: s00013-012-0424-9
OpenAlex ID: W2087779510
cites work: Vector bundles trivialized by proper morphisms and the fundamental group scheme
cites work: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin
cites work: Detecting infinitely many semisimple representations in a fixed finite dimension.
cites work: Q3765876
cites work: Some remarks on the local fundamental group scheme
cites work: On the fundamental group scheme
cites work: The fundamental group-scheme
cites work: Frobenius-unstable bundles and $p$-curvature
cites work: A smooth counterexample to Nori’s conjecture on the fundamental group scheme
cites work: Algebraic Function Fields and Codes
DOI: 10.1007/S00013-012-0424-9
description: scientific article; zbMATH DE number 6109278