  Hankel spectrahedra are the dual convex cones to the cone of sums of squares
of real polynomials, and we study them from the point of view of convex
algebraic geometry. We show that the Zariski closure of the union of all
extreme rays of Hankel spectrahedra for ternary forms is an irreducible variety
of codimension 10. It is the variety of all Hankel (or middle Catalecticant)
matrices of corank at least 4. We explicitly construct a rational extreme ray
of maximal rank using the Cayley-Bacharach Theorem for plane curves. We work
out the rank stratification of the semi-algebraic set of extreme rays of Hankel
spectrahedra in the first three nontrivial cases d = 3, 4, 5. Dually, we get a
characterisation of the algebraic boundary of the cone of sums of squares via
projective duality theory, extending previous work of Blekherman, Hauenstein,
Ottem, Ranestad, and Sturmfels.
