Clifford Indices for Vector Bundles on Curves

Abstract

For smooth projective curves of genus g ≥ 4, the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices.

On bundles of rank 3 computing Clifford indices

“…The idea of generalising the classical Clifford index Cliff(C) to higher rank vector bundles was proposed some 20 years ago, but formal definitions and the development of a basic theory took place much more recently [11]. 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].…”

“…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. In the present paper, we improve the lower bounds of [12] in various circumstances.…”

Vector Bundles of Rank 2 Computing Clifford Indices

“…In discussing γ ′ 2 , we shall make much use of the Lemma of Paranjape and Ramanan [14,Lemma 3.9] (see also [8,Lemma 4.8]), which we now state for the case of bundles of rank 2.…”

“…In a previous paper [8], we introduced Clifford indices for vector bundles, generalising the classical definition, as follows. For any vector bundle E of rank r E and degree d E on C, consider…”

“…If in addition γ(E) = γ n (respectively γ(E) = γ ′ n ), we say that E computes γ n (respectively γ ′ n ). Our object in the present paper is to study the bundles which compute γ 2 or γ We have always d r < d r+1 and d r+s ≤ d r + d s ; in particular d n ≤ nd 1 for all n (see [8,Section 4]). We say that d r computes γ 1 if d r ≤ g − 1 and d r −2r = γ 1 and that C has Clifford dimension r if r is the smallest integer for which d r computes γ 1 .…”