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

Extreme Rays of the Hankel Spectrahedra for Ternary Forms