= Genus-two quartic from a degree-four complete linear system
On a smooth projective curve of genus two, $D=K+P+Q$ has three sections and no base points by the <Riemann-Roch theorem>. Its map to $\mathbb P^2$ is nondegenerate and has mapping degree times image degree four. The only nonbirational possibility is a double cover of a conic. Every degree-two divisor with two sections is canonical, again by <Riemann-Roch theorem>, so that possibility forces $D\sim2K$, or $P+Q\sim K$. If this equivalence fails, the image is a quartic and the map is birational. The quartic is singular because a smooth plane quartic has genus three.
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