Gonality and Clifford index of curves on elliptic \(K3\) surfaces with Picard number two (Q267051)

Let \(X\) be a \(K3\) surface with Picard number 2, and such that the Picard group of \(X\) is generated by two curves \(E,F\) with selfintersection \(0\). Let \(m\) be the intersection number of \(E\) and \(F\). Let \(C\) be a curve on \(X\) of genus \(g>2\). The author shows that one of the following cases happens: (I) The Clifford index of \(C\) is cut out by an elliptic curve \(E_C\). The curve \(E_C\) is linearly equivalent with the one among \(E\) or \(F\) with the smallest intersection number with \(C\). The Clifford index of \(C\) is \(C . E_C-2\) and \(C.E_C\) is the gonality of \(C\) (II) We have \(m>2\) and \(C\) is linearly equivalent with \(E+F\). Then \(C\) has maximal Clifford index \(\lfloor m/2 \rfloor\).