Let \(f : X \to C\) be a branched covering of degree \(k\) of a smooth complex curve \(C\) of genus \(q \geq 1\); let \(g\) be the genus of \(X\). Here we study the linear series on \(X\).
First, if \(k = 2\) and \(g \geq 5q + 1\) we prove that for every \(d \geq g - q\) there is a base point free \(g^1_d\) not composed with the involution \(f\). – Then, for every \(C\) but with \(f\) general we prove that if \(d \geq kq - k + 3\) every irreducible component of \(W^1_d (X)\) containing a base point free \(g^1_d\) inducing with \(f\) a birational morphism \(X \to C \times \mathbb{P}^1\) is generically smooth of dimension \(\rho (d,g,1) : = 2d - 2 - g\); the proof uses the deformation theory of curves contained in \(C \times \mathbb{P}^1\); if \(2g \geq 2 k^2 - (2k - 3) (k - 1)\) we may avoid the birationality assumption by Castelnuovo-Severi inequality. If both \(C\) and \(f\) are general and \(kq \leq g \leq kq + (k - 1)\), then we prove that \(X\) satisfies the full theorem of Gieseker – Petri; the proof uses the theory of limit linear series.