On an Example of Mukai

Abstract

Abstract. In this paper we use an example of Mukai to construct semistable bundles of rank 3 with six independent sections on a general curve of genus 9 or 11 with Clifford index strictly less than the Clifford index of the curve. The example also allows us to show the non-emptiness of some Brill-Noether loci with negative expected dimension.

On bundles of rank 3 computing Clifford indices

“…For n = 3 it fails for the general curve of genus 9 or 11 (see [10]) and for curves of genus ≥ 12 contained in K3 surfaces [8,Corollary 1.6]. For n = 2 it is still conjectured to hold for the general curve of any genus (see [7]).…”

“…Hence h 0 (M) = 3 and F ≃ E M . The semistability of E now follows as in [10,Proposition 3.5] noting that the inequality 3d 1 ≥ 2d 2 is weaker than 3 Cliff(C) ≥ 2d 2 −6. Moreover, E is obviously generated.…”

“…In the present paper, we improve the lower bounds of [12] in various circumstances. As a result, we are able to compute values of Cliff 3 (C) in some cases and to give examples for which Cliff 3 (C) > Cliff 2 (C), thus answering in the affirmative Question 5.7 in [10].…”

“…Examples are also known for g = 9 and g ≥ 11 for which the rank-3 Clifford index Cliff 3 (C) is strictly smaller than Cliff(C) [10,8] and lower bounds for Cliff 3 (C) had previously been established in [12]. However, with the exception of the case where Cliff(C) ≤ 2 (when Cliff 3 (C) = Cliff(C) [11,Proposition 3.5]), no actual values of Cliff 3 (C) are known.…”

Twists of Mukai bundles and the geometry of the level $3$ modular variety over $\overline {\mathcal {M}}_{8}$

“…Analogously to their counterparts in genus 6 and 8 their global sections give embeddings of the curve, albeit in an orthogonal or a symplectic Grassmannian. These bundles too exhibit interesting properties: for instance on a general genus 9 curve they were used to give early counterexamples to Mercat's conjecture (see [LMN12]). …”