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Covering of curves, gonality, and scrollar invariants. (English) Zbl 1142.14018

Let \(f:X\to Y\) be a \(k\)-covering between two (connected, non-singular, complete) curves of genus \(g\) and \(q\) respectively. By Riemann–Hurwitz, \(g-1\geq k(q-1)\). If \(g\) is large enough, Castelnuovo–Severi inequality implies that any rational function \(u:X\to \mathbb{P}^1\) such that \(\operatorname{deg}(u)=\operatorname{gon}(X)\) factors through \(f\). Theorem 1 shows the existence of examples which do not satisfy this property; moreover, under certain conditions, \(Y\) can be of general moduli. On the other hand, if \(f\) does not factor non-trivially through another smooth curve, Castelnuovo–Severi inequality implies the following cohomological restriction on the associated bundle \(E_f:=f_*((\mathcal O_X)/\mathcal O_Y\) (Theorem 2): there exists no effective divisor \(D\) on \(Y\) such that \(\operatorname{deg}(D)\leq (g-kq)/k(k-1\), \(h^0(Y,{\mathcal O}(D))=1\) and \(h^0(Y,E_f(D))>0\).

MSC:

14H51 Special divisors on curves (gonality, Brill-Noether theory)
14H50 Plane and space curves
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