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Clifford index of complete intersections of space. (Indice de Clifford des intersections complètes de l’espace.) (French) Zbl 0871.14027

Let \(C\) be an algebraic curve. We define as gonality of \(C\) the smallest integer \(n\) such that there exists a morphism \(C\to \mathbb{P}^1\) of degree \(n\); i.e. the smallest integer \(n\) such that there exists an invertible sheaf \({\mathcal L}\) on \(C\) of degree \(n\) with \(h^0 ({\mathcal L}) \geq 2\). We define as Clifford index of an invertible sheaf \({\mathcal L}\) on \(C\) the integer \(\deg ({\mathcal L}) -2[h^0 ({\mathcal L})-1]\) and we define as Clifford index of \(C\) the smallest Clifford index of an invertible sheaf \({\mathcal L}\) satisfying \(h^0 ({\mathcal L}) \geq 2\) and \(h^0 ({\mathcal L}^\vee \otimes \omega_C) \geq 2\).
Let \(C\subset \mathbb{P}^3\) be a twisted smooth complete intersection curve and let \(\ell\) be the maximum degree of a collinear positive divisor on \(C\). In the paper under review the author shows that the gonality of \(C\) is \(\deg (C)-\ell\) and that an effective divisor \(\Gamma\) on \(C\) computes this gonality if and only if \(\Gamma\) is the residual of a collinear divisor of degree \(\ell\) on a plane section of \(C\).
Moreover, by results of G. Martens and others [cf., M. Coppens and G. Martens, Compos. Math. 78, No. 2, 193-212 (1991; Zbl 0741.14035)], the author shows that, if \(\deg (C)\neq 9\), the Clifford index of \(C\) is \(\deg (C)-\ell -2= \text{gon} (C)-2\).

MSC:

14H50 Plane and space curves
14M10 Complete intersections
14N05 Projective techniques in algebraic geometry

Citations:

Zbl 0741.14035
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References:

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