Let $\mathcal{C}$ be a (smooth, complex, projective) genus 2 curve. Take two different points $p,q\in\mathcal{C}$ and let $K$ be the canonical divisor class. I know (by means of Riemann-Roch) that the divisor $K+p+q$ defines a map $\mathcal{C}\rightarrow\mathbb{P}^2$ whose image (if I'm correct) is a curve with just one node.

A couple of questions: Is this a birational map? How can I visualize in a more concrete geometrical way? I suffer lack of concrete examples of many things happening in Algebraic Geometry so I will appreciate if somebody can tell me why this should be an interesting one.


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