A counter example of Arrondo-Sols' conjecture
Arrondo-Sol's conjecture can be stated as follows: Let \(E\) be a rank-2 vector bundle of degree \(d\) on a smooth complex algebraic curve of genus \(g\), \(-e\) be the minimal self-intersection number of a unisecant curve in the ruled surface \(p(E)\), and \(r+1=h^ 0(E)\). If \(-e \leq d \leq 4g-4+e\) and \(E \neq L \oplus L\) \((L\) line subbundle), then \(r \leq(d+e)/2\) [cf. \textit{E. Arrondo} and \textit{I. Sols} in Complex projective geometry, Sel. Pap. Conf. Proj. Var., Trieste 1989, and Vector Bundles and Special Proj. Embeddings, Bergen 1989, Lond. Math. Soc. Lect. Note Ser. 179, 24-31 (1992; Zbl 0772.14012)]. -- If the curve is hyperelliptic, then the conjecture is true (loc. cit.). The paper under review shows, by an example, that the conjecture is false for nonhyperelliptic curves of genus 3.
scientific article; zbMATH DE number 109357