I was reading Beauville's Complex algebraic surfaces, at page 5 there is an example in which curves in $\mathbb{P}^1 \times \mathbb{P}^1$ are classified by the bidegree up to linear equivalence.
I'm happy with that, and it makes sense that the defining polynomial determines the linear equivalence class of the curve. However, the bidegree is also the degree of the morphism defined by the projections to the lines $\lbrace 0 \rbrace \times \mathbb{P}^1$ and $\mathbb{P}^1 \times \lbrace 0 \rbrace$ . Thence, I was wondering if the bidegree of the curve also gives some contraints on the other possible morphisms from the curve to the two lines above.
I heard somewhere that the degree of the curve is somehow related to its gonality, for instance, is there such a relation?
I'm sorry if my question can sound confused, I just started facing algebraic geometry topics and still don't know precisely how to work with these concepts practically.