Hyperelliptic curves among cyclic coverings of the projective line, I

Cyclic covering of the projective line with prime gonality

“…By (3.2), we have w ≤ n − 2. By [4,Theorem 1], if w = n − 2, then V is elliptic or hyperelliptic. Thus, we may assume that w ≤ d − 3.…”

“…In this note, we denote the gonality of V by Gon(V ). The condition for Gon(V ) = 1 is trivial, and the condition for Gon(V ) = 2 has been given in [3] and [4]. Thus, we may always assume that Gon(V ) ≥ 3 in this note.…”

The slope of fibred surfaces: Unitary rank and Clifford index

“…Moreover, since the general fibre 𝐹 has equation (5.1), applying the results in [42,43] we conclude that 𝐹 has maximal gonality 4, and by Ballico's result Cliff(𝐹) = gon(𝐹) − 2 = 2. Observe moreover that the general fibre is not bielliptic, as can be derived for instance by [40,Lemma 2.4], so the results of [6] By direct computations, one can see in this case that we have 𝐾 2 𝑓 ∕𝜒 𝑓 = 45∕4, which is strictly greater than 4.…”

The slope of fibred surfaces: unitary rank and Clifford index

“…Moreover, since the general fibre F has equation 5.1, applying the results in [38] and [39], we conclude that F has maximal gonality 4 and -by Ballico's result-Cliff(F ) = gon(F ) − 2 = 2. Observe moreover that the general fibre is not bielliptic as can be derived for instance by [37,Lemma 2.4], so the results of [5] do not apply.…”