Cohomological properties of multiple coverings of smooth projective curves (Q1880834)

Let \(f:\mathcal X\to\mathcal C\) be a morphism of degree \(k\) between smooth connected projective curves of genus \(g\) and \(q\) respectively. In this paper the study of cohomological properties of the sheaf \(E:={f_*}(\mathcal O_\mathcal X)/\mathcal O_\mathcal C\) is considered. If \(q=0\), \(E\) is a direct sum of \(k-1\) line bundles, and the rank 1 summands of \(E\) uniquely determine the scrollar invariant of \(f\) [see \textit{F. O. Schreyer}, Math. Ann. 275, 105--137 (1986; Zbl 0578.14002)]. In this paper, the author considers the analogous problem for \(q>0\). He also introduces the notion of a (total) Weierstrass point of \(f\) at \(P\in\mathcal X\) as follows. Let \(N(f,P)\) and \(n(f,P)\) denote respectively the sequence of integers \(h^0(C,f_*(\mathcal O_\mathcal X)(tP))\) and \(h^0(C,E(tP))\), \(t\in \mathbb N_0\). The point \(P\) is called a total Weierstrass point (Weierstrass point) if \(N(f,P)\neq N(f,Q)\) (\(n(f,P)\neq n(f,Q)\)) for a generic point \(Q\in \mathcal X\). The author presents several examples to illustrate his results.