Gonality and Clifford index of projective curves on ruled surfaces

Abstract

Let X be a smooth curve on a ruled surface π : S → C. In this paper, we deal with the questions on the gonality and the Clifford index of X and on the composedness of line bundles on X with the covering morphism π| X. The main theorem shows that if a smooth curve X ∼ aC o + bf satisfies some conditions on the degree of b, then a line bundle L on X with Cliff(L) ≤ ag(C) − 1 is composed with π| X. This implies that a part of the gonality sequence of X is computed by the gonality sequence of C as follows: d r (X) = ad r (C) for r ≤ L, where L is the length of the gonality sequence of C.

A note on gonality of curves on general hypersurfaces

“…For all admissible cases, a family of expected dimension of nodal curves in |O X (1)| with geometric genus g, carrying a g 1 k on their normalizations, has been exhibited. Similar problems have been investigated in different settings, for instance on rational and ruled surfaces [3,10,37,48], Enriques surfaces [31,32], elliptic K3's [46], toric surfaces [29], as well as abelian ones [30,44].…”

Gonality and genus of canonical components of character varieties

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