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On the gonality sequence of an algebraic curve. (English) Zbl 1238.14022

The gonality sequence \(d_1,d_2,\dots,d_r,\dots\) of a projective curve \(C\) is defined by setting \(d_r=\) the smallest integer \(i\) for which one has a nondegenerate rational map \(C\to\mathbb P^r\), of degree \(i\). For many classes of curves, the gonality sequence satisfies the slope inequalities \[ \frac {d_r}{r} \geq \frac{d_{r+1}}{r+1}. \] However, there are cases in which some of the previous inequalities are violated. For instance, they are violated by plane curves of degree \(n\geq 5\), for some values of \(r\).
The authors determine more classes of curves for which the previous inequalities are violated. They find that the \(r\)-th slope inequalities are violated in the following cases:
- extremal curves of degree \(n\geq 3r-1\) in \(\mathbb P^r\), \(r\geq 2\);
- smooth curves of degree \(n\geq 8(r-1)\) on a general \(K3\) surface of degree \(2r-2\) in \(\mathbb P^r\), \(r\geq 3\);
- smooth complete intersection of type \((s,p)\) in \(\mathbb P^3\), with \(2\leq s\leq p\;\) and \(p\geq 4\).

MSC:

14H51 Special divisors on curves (gonality, Brill-Noether theory)
14H45 Special algebraic curves and curves of low genus

Keywords:

curves
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References:

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