The Clifford index of line bundles on a 2-elementary K3 surface given by a double cover of a Del Pezzo surface

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We call a K3 surface \(X\) a 2-elementary K3 surface if the Neron–Severi lattice \(S_X\) satisfies the condition that \(S_X^{*}/S_X\) is a 2-elementary group, where \(S_X^{*}:={{\mathrm{Hom}}}(S_X,\mathbb {Z})\) . In this paper, in the case where \(X\) is a 2-elementary K3 surface given by a double cover of a smooth Del Pezzo surface of degree \(4\le d\le 8\) and \(L\) is a base point free and big line bundle on \(X\), for any smooth curve \(C\) in the linear system \(|L|\), we investigate line bundles on \(C\) which compute the Clifford index of \(C\).