Models for elliptic surfaces with a bisection

In the study of (relatively minimal) elliptic surfaces $f\colon S \to C$ with a section, the Weierstrass model plays an important role. We discuss two different models for elliptic surfaces with a bisection (or more generally an $f$-nef divisor which has degree two on the fibres): a birational model induced by the involution on the general fibre and a relative determinantal model coming from the relative section ring. The presence of multiple fibres gives rise to new phenomena, which will be illustrated for Enriques surfaces or rational elliptic surfaces. (joint work with Anna Ulivi and Aline Zanardini)