Non-negative polynomials on generalized elliptic curves

We study the cone of non-negative polynomials on generalized elliptic curves. We show that the zero set of every extreme ray has dense real points. If a generalized elliptic curve is embedded via a complete linear system, then we show that the convex hull of its real points (taken inside any affine chart containing all real points) is a spectrahedron. On the way, we generalize a result by Geyer–Martens on 2-torsion points in the Picard group of smooth real curves (of arbitrary genus) to possibly singular and reducible ones.