On the gonality sequence of an algebraic curve

Abstract

For any smooth irreducible projective curve X, the gonality sequence {d r | r ∈ N} is a strictly increasing sequence of positive integer invariants of X. In most known cases d r+1 is not much bigger than d r . In our terminology this means the numbers d r satisfy the slope inequality. It is the aim of this paper to study cases when this is not true. We give examples for this of extremal curves in P r , for curves on a general K3-surface in P r and for complete intersections in P 3 .

On the gonality sequence of smooth curves

“…Hence C does not satisfy the slope inequality if and only if there is at least one integer r ≥ 2 for which (1.1) fails. Many different examples of such curves are constructed in [7]. As a byproduct of our study of nodal plane curves we get the following result, which answers [7,Question 5.4].…”

“…For each integer r ≥ 1 the r-gonality d r (C) of C is the minimal integer d such that there is a degree d line bundle L on C such that h 0 (C, L) ≥ r + 1 [7]. The sequence {d r (C)} r≥1 is called the gonality sequence of C. This sequence is important to understand the Brill-Noether theory of vector bundles on C [8][9][10].…”

“…The sequence {d r (C)} r≥1 is called the gonality sequence of C. This sequence is important to understand the Brill-Noether theory of vector bundles on C [8][9][10]. See [7,Section 3], for general properties of this sequence for an arbitrary curve C. For most curves we have…”

“…In [7] H. Lange and G. Martens introduced the following notion. C is said to satisfy the slope inequality if (1.1) is satisfied for all r ≥ 2.…”

On the birational gonalities of smooth curves

“…Equality holds if d r (C) < r · d 1 (C) and C has no non-trivial morphism onto a smooth curve of positive genus. In [6] H. Lange and G. Martens studied the slope inequality for the usual gonality sequence of smooth curves (it may fail for some C, but not for a general C).…”