Vector Bundles of Rank 2 Computing Clifford Indices

Abstract

Abstract. Clifford indices of vector bundles on algebraic curves were introduced in a previous paper of the authors. In this paper we study bundles of rank 2 which compute these Clifford indices. This is of particular interest in the light of recently discovered counterexamples to a conjecture of Mercat.

On bundles of rank 3 computing Clifford indices

“…In fact, for n = 2, this form of the conjecture is equivalent to the original (see [15,Proposition 2.7]). Lemma 2.1.…”

“…However, the conjecture is known to fail in many other cases (see [7,8,14,15,16]). 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].…”

“…(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.…”

“…Since then, there have been major developments, in particular the construction of curves for which the rank-2 Clifford index Cliff 2 (C) is strictly less than Cliff(C) [7,8,14,15,16], thus producing counter-examples to a conjecture of Mercat [18]. A good deal is now known about bundles computing Cliff 2 (C) [15].…”

Scroll codes over curves of higher genus: reducible and superstable vector bundles

“…The conjectural bound in [11, p. 611], depends on a conjecture of Mercat [20], which is in general false [ [14][15][16][17][18]. We think that in the long run a very weak form of it will be proved (even in positive characteristic).…”