Vague rambling: I hate asking these types of questions, but I feel that I would benefit immensely from hearing some discussion of the use of hyperelliptic curves in constructing certain examples. What do I mean? Of course, as mathematicians, and especially as (aspiring) algebraic geometers, either while reading, examples are essential. Often times, however, hyperelliptic curves are mentioned. They are mentioned so often (from what I see) that they must have some property that is very useful when trying to construct examples.
Let us consider an actual instance of this: Following Campana's survey on special manifolds, define a Bogomolov sheaf to be a saturated, coherent subsheaf $\mathcal{L} \subset \Omega_X^p$ of rank one over a compact Kähler manifold $X$ such that the Kodaira dimension $\kappa(X, \mathcal{L})=p>0$. The manifold $X$ is said to be special if $X$ admits no Bogomolov sheaves.
We'll construct an example of a Bogomolov sheaf as follows: Let $E$ be an elliptic curve and $\tau$ a translation of order $2$ on $E$. Let $C$ be a hyperelliptic curve with an involution $\eta$. Set $X : = (E \times C)/\langle \tau, \eta \rangle$ and let $f : X \to B$ be the Moishezon--Iitaka map onto some curve $B$. The saturation of $f^{\ast}K_B$ in $\Omega_X^1$ has Kodaira dimension $\kappa(X, f^{\ast}K_B)=1$ and is therefore a Bogomolov sheaf. In fact, it is the only one on $X$.
Question: What properties do hyperelliptic curves have that make them useful in the construction of examples?
Feel free to downvote my question if this question is better suited for math.stackexchange. The reason for posting the problem to overflow is that the motivation is clearly at the level of research mathematics. And the question is, fundamentally, one of motivation.
