Clifford Indices Research Articles

Higher rank Clifford indices of curves on a K3 surface

Let (X, H) be a polarized K3 surface with mathrm {Pic}(X) = mathbb {Z}H, and let Cin |H| be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when gge r^2ge 4, the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree d(ge 5) smooth plane curve is d-4, which is the same as the Clifford index of the curve.

Bundles computing Clifford indices on trigonal curves

In this paper, we determine bundles which compute the higher Clifford indices for trigonal curves.

Vector Bundles of Rank 2 Computing Clifford Indices

Clifford indices of vector bundles on algebraic curves were introduced in a previous article of the authors. In this article 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.

STABLE BUNDLES OF RANK 2 WITH FOUR SECTIONS

This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve C. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve C of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems. The results can also be expressed in terms of Koszul cohomology and the methods provide a useful tool for the study of the geometry of the moduli space of curves.

Lower bounds for Clifford indices in rank three

AbstractClifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers by the authors. In this paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.

Generation of vector bundles computing Clifford indices

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.

An inequality of Clifford indices for a finite covering of curves

We prove that for a finite covering of curves the Clifford index of the source is at least that of the target.

Clifford indices of ribbons

We present a theory of limit for rational ribbonsthat is, for schemes that are double structures on PI . This allows us to define a linear series Clifford for ribbons. Our main theorem shows that this is the same as the Clifford index of ribbons studied by Eisenbud-Bayer in this same volume. This allows us to prove that the Clifford index is semicontinuous in degenerations from a smooth curve to a ribbon. A result of Fong [ 1993] then shows that ribbons may be deformed to smooth curves of the same Clifford index. Thus the Canonical Curve Conjecture of Green [1984] would follow, at least for a general smooth curve of each Clifford index, from the corresponding statement for ribbons.

Remarks on Clifford indices.

Remarks on Clifford indices.