On strictly elliptic K3 surfaces and del Pezzo surfaces

This article primarily aims at classifying, on certain K3 surfaces, the elliptic fibrations induced by conic bundles on smooth del Pezzo surfaces. The key geometric tool employed is the Alexeev-Nikulin correspondence between del Pezzo surfaces with log-terminal singularities of Gorenstein index two and K3 surfaces with non-symplectic involutions of elliptic type: the latter surfaces are realized as appropriate double covers obtained from the former ones. The main application of this correspondence is in the study of linear systems that induce elliptic fibrations on K3 surfaces admitting a strictly elliptic non-symplectic involution, i.e., whose fixed locus consists of a single curve of genus $g\geq 2$. The obtained results are similar to those achieved by Garbagnati and Salgado for jacobian elliptic fibrations.
Comment: v2: 21 pages. Corrections in Table 1. Comments are very welcome!