Let \(X\) be a smooth connected projective curve defined over \(\mathbb{R}\). The real gonality, \(\text{gon} (X,\mathbb{R})\) of \(X\) is the minimal integer \(k\) such that there is \(L\in\text{Pic}(X)\) with \(\deg(L)=k\), \(h^0(X,L) =2\) and \(L\) defined over \(\mathbb{R}\). Here we give bounds for the real gonality of \(X\) in terms of the complex gonality of \(X\). When \(X(\mathbb{R})= \emptyset\) strange phenomena may occur:
Here we give an example of a real curve \(X\) with a real line bundle \(L\) such that \(h^0(X,L) >2\) and \(\deg(L)< \text{gon}(X,\mathbb{R})\).