A bielliptic surface is a surface of type $S=E\_1 \times E\_2/G$ where $E\_1, E\_2$ are elliptic curves and $G$ is a finite group of translations of $E\_1$ acting on $E\_2$ such that $E\_2/G=\mathbf{P}^1$.
It is easy to classify these surfaces, and it turns out that there are $7$ families.


Is there a classification of the base point free and of the very ample linear systems on a bielliptic surface?



Linear systems on bielliptic surfaces
The structure of $\textrm{Num}(S)$ for a bielliptic surface $S$ is given in the paper by F. Serrano [*Divisors of bielliptic surfaces and embeddings in* $\mathbb{P}^4$](http://link.springer.com/article/10.1007/BF02570754), Mathematische Zeitschrift **203** (1990), 527-533. 


First, Serrano proves that a basis for $\textrm{Num}(S) \otimes \_{\mathbb{Z}} \mathbb{Q} \cong H^2(S, \, \mathbb{Q})$ is given by the fibres $A$and $B$ of the two natural, isotrivial elliptic fibrations $f\_1\colon S \to E\_1/G$ and $f\_2 \colon S \to E\_2/G$. They clearly satisfy
$$A^2=B^2=0, \quad AB = |G|.$$ 


This allows the author to describe the ample cone of $S$: in fact, it follows by Nakai-Moishezon criterion that a divisor in the numerical class $\alpha A + \beta B$ is ample if and only if $ \alpha >0$ and $\beta >0$, see Lemma 1.3.


An explicit description of a basis of $\textrm{Num}(S)$ over $\mathbb{Z}$ must take into account the multiplicities of the singular fibres of $f\_1$ and $f\_2$. These depend on which of the seven families we are considering, and a complete case-by-case analysis is given in Theorem 1.4.


Finally, some results on *very* ample divisors are given in Theorem 2.1: in fact, Serrano shows that the bielliptic surfaces in one of the seven families admit a very ample linear system $|L|$ such that $H^0(S, \,L)=5$ and $L^2=10$. Consequently, these surfaces can be embedded in $\mathbb{P}^4$ as surfaces of degree $10$. 


