Generation of vector bundles computing Clifford indices

Abstract

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least four were defined in a previous paper of the authors. The present paper studies bundles which compute these Clifford indices. We show that under certain conditions on the curve all such bundles and their Serre duals are generated.

On bundles of rank 3 computing Clifford indices

“…Under the same hypotheses as those of Proposition 7.1, it was proved in [13] that E is generated. We therefore have an evaluation sequence…”

“…(this invariant is denoted in [11,12,13,14,15] by γ ′ n ). Note that Cliff 1 (C) = Cliff(C) is the usual Clifford index of the curve C. We say that E contributes to Cliff n (C) if E is semistable of rank n with h 0 (E) ≥ 2n and µ(E) ≤ g − 1.…”

On an Example of Mukai

“…where µ(E) = d n . Then the Clifford index Cliff n (C) is defined by [6,7,8,9,10]). We say that a bundle E contributes to Cliff n (C) if it is semistable of rank n with µ(E) ≤ g −1 and h 0 (E) ≥ 2n and that E computes Cliff n (C) if in addition γ(E) = Cliff n (C).…”

Bundles computing Clifford indices on trigonal curves

“…Since h 0 (M 2 (−p)) = 1 and E is generated by [6,Theorem 2.4], this must factor as E → O C → M 2 (−p). But h 0 (E * ) = 0, since E is semistable.…”