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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?

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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 $\mathbb{P}^4$, 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 $10$.

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Neat, so they must all be realizable as Horrocks-Mumford surfaces. Is that how Serrano describes them? –  Noam D. Elkies Jul 31 at 14:01
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@NoamD.Elkies: It seems to me that Horrocks-Mumford surfaces are abelian surfaces, not bielliptic ones. Am I missing something? –  Francesco Polizzi Jul 31 at 14:18
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In fact, in the introduction of his paper Serrano says that these bielliptic surfaces provide the third known family of surfaces of degree $10$ in $\mathbb{P}^4$, besides Horrock-Mumford surfaces and elliptic quintic scrolls. –  Francesco Polizzi Jul 31 at 14:22
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There are many family of surfaces of degree 10 in $\Bbb{P}^4$ (rational, K3...), they have been classified by Ranestad. The point is that the three families you mention consist of irregular surfaces ($q\geq 1$). To my knowledge they are still the only known (smooth) irregular surfaces in $\Bbb{P}^4$. –  abx Jul 31 at 16:37
    
@abx: You are right, I forgot to write "irregular" –  Francesco Polizzi Jul 31 at 17:24

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