Projective surfaces with bi-elliptic hyperplane sections (Q1814202)

Let \(X\) be a projective smooth surface which possesses some bi-elliptic curves among its hyperplane sections \(| L|\), i.e. non- hyperelliptic curves \(C\in| L|\), with genus \(g(C)\geq 3\) which are double coverings of elliptic curves (these curves are also known as superelliptic). In the paper a complete characterization of those surfaces of degree \(d\geq 17\) and \(d\leq 8\) is given. The main result is that, if \(d\geq 19\), either \((X,L)\) is a scroll over a bi-elliptic curve, or \((X,L)\) is an elliptic conic bundle. Examples and some results of independent interest about special linear series on bi-elliptic curves are also given. The main tool used is the study of the adjunction mapping on \(X\). Other techniques related to Reider's method are also used.