The gonality and the Clifford index of curves on an elliptic ruled surface
Let \(S\) be a geometrically ruled surface (i.e. a \(\mathbb P^1\)-bundle) over an elliptic curve \(C\) and \(X \subset S\) a smooth curve. Here the author for almost all numerical classes is able to explicitly compute the gonality and the Clifford index of \(X\). For the case of \(\mathbb P^1\)-bundles over \(\mathbb P^1\) (even for singular curves), see [\textit{G. Martens}, Arch. Math. 67, 349--352 (1996; Zbl 0872.14022)]. In most cases these invariants are computed using the ruling of \(S\), but the author found several exceptional cases.
\(\mathbb P^1\)-bundle over an elliptic curve
scientific article; zbMATH DE number 2155975