On an example of Mukai (Q2882491)

Let \(C\) be a smooth projective curve of genus \(g\geq 4\) over an algebraically closed field of characterictic zero, \(\mathrm{Cliff}(C)\) its classical Clifford index, and \(\mathrm{Cliff}_n(C)\) its \(n\)th Clifford index introduced by {H. Lange} and \textit{P. E. Newstead} [in: Affine flag manifolds and principal bundles. Basel: Birkhäuser, 165--202 (2010; Zbl 1227.14038)]. The main purpose of the paper is to give examples when \(\mathrm{Cliff}_n(C) \neq \mathrm{Cliff}(C)\). For this purpose authors use a bundle of rank 3 on a curve of genus 9 with 6 independent sections constructed by \textit{S. Mukai} [Proc. Japan Acad., Ser. A 68, No. 1, 7--10 (1992; Zbl 0768.14014); Ann. Math. (2) 172, No. 3, 1539--1558 (2010; Zbl 1210.14034)]. Also they show that the construction works for genus 11, and consider possible generalisations and extensions. In the main theorem, general conditions are given under which \(\mathrm{Cliff}_n(C)< \mathrm{Cliff}(C)\) either for \(n=2\) or \(n=3\). The constructed examples prove non-emptiness of corresponding Brill-Noether loci. The paper is concluded with list of open questions and observations on possible answers.