Extreme Rays of the Hankel Spectrahedra for Ternary Forms
Grigoriy Blekherman
Rainer Sinn
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.
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\begin{document}
\title{Extreme Rays of the Hankel Spectrahedra for Ternary Forms}
\author{Grigoriy Blekherman}
\address{Georgia Institute of Technology, Atlanta, Georgia, USA}
\email{greg@math.gatech.edu}
\author{Rainer Sinn}
\address{Georgia Institute of Technology, Atlanta, Georgia, USA}
\email{sinn@math.gatech.edu}
\subjclass[2010]{Primary: 13P25, 14P10, 05E40; Secondary: 14N99}
\keywords{sums of squares, non-negative polynomials, Hankel matrices, Cayley-Bacharach, spectrahedra, apolar ideals}

\begin{abstract}
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.
\end{abstract}

\maketitle

\section{Introduction}
In polynomial optimisation it is a central problem to certify that a real polynomial in $n$ variables is non-negative as a function on $\R^n$. One prominent approach is to write it as a sum of squares of polynomials which is studied in real algebraic geometry and used in semi-definite programming methods, cf.~\cite{BleParThoMR3075433}.

This idea goes back to work by Hilbert \cite{HilMR1510517}, where he proved that every non-negative polynomial is a sum of squares in $1$ variable, in degree $2$ and for bivariate polynomials of degree $4$. In every other case, there are non-negative polynomials that cannot be written as a sum of squares of polynomials. Hilbert's proof relies on methods from algebraic geometry, in particular the Cayley-Bacharach Theorem. In modern words, Hilbert constructed a supporting hyperplane to the sums of squares cone in a strictly positive polynomial, i.e.~a linear functional that is non-negative on all sums of squares but takes a negative value in a strictly positive polynomial, see \cite{Reznick}.
We revisit this construction using modern developments in classical algebraic geometry with the aim of controlling the rank of the associated Hankel matrix. The most important ingredients are the Cayley-Bacharach Theorem and the Buchsbaum-Eisenbud Structure Theorem for height $3$ Gorenstein algebras. Here is our setup:

A linear functional $\ell$ on the real vector space of ternary forms of degree $2d$ is non-negative on every square if and only if the bilinear form 
\[
B_\ell\colon \left \{
\begin{array}[]{rcl}
\R[x,y,z]_d\times \R[x,y,z]_d & \to & \R \\
(f,g) & \mapsto & \ell(f\cdot g).
\end{array}
\right.
\]
is positive semi-definite. The representing matrix of this bilinear form with respect to the monomial basis is the Hankel matrix associated with $\ell$. 
Therefore, the convex cone dual to the cone $\Sigma_{2d}$ of sums of squares of polynomials is the Hankel spectrahedron
\[
\Sigma_{2d}^\vee = \{\ell\in\R[x,y,z]_{2d}^\ast \colon (\ell(x^{\alpha+\beta}))_{\alpha,\beta} \text{ is positive semi-definite}\}.
\]

Every real point evaluation $\ev_x\colon\R[x,y,z]_{2d}\to \R$, $p\mapsto p(x)$, at $x\in\R^3$ is an extreme ray of $\Sigma_{2d}^\vee$. In fact, by the Veronese embedding of $\PP^2$ of degree $2d$, they are exactly the positive semi-definite rank $1$ Hankel matrices. We are interested in extreme rays of higher rank. These correspond to supporting hyperplanes of $\Sigma_{2d}$ which expose a face whose relative interior consists of strictly positive polynomials. Conversely, for every non-negative polynomial $p$ that is not a sum of squares, there exists an extreme ray $\R_+\ell$ of $\Sigma_{2d}^\vee$ such that $\ell(p)<0$. 

Our construction of extreme rays of higher rank is a generalisation of a construction of the first author, which extends Hilbert's methods and characterises the extreme rays of $\Sigma_6^\vee$ of rank greater than $1$ by special point configurations in the plane, namely the transversal intersection of two plane cubics in $9$ points, see \cite{BleNP}. 
In our main result, we construct an extreme ray $\R_+ \ell$ of $\Sigma_{2d}^\vee$ for $2d\geq 8$ such that the associated bilinear form $B_\ell$ has maximal rank, which is $\binom{d+2}{2}-4$ by Theorem \ref{Thm:MaxRank}. 
Using the extensive work on height $3$ Gorenstein algebras, we conclude that there are many extreme rays of maximal rank; precisely, the Zariski closure of the set of extreme rays is an irreducible variety of codimension $10$. The intricate rank stratification of the semi-algebraic set of extreme rays characterises the algebraic boundary of the sums of squares cone via projective duality theory. We completely work out the first three nontrivial cases $d=3,4,5$ in Section \ref{sec:Dextics}, extending the study of the algebraic boundary of the sums of squares cones for ternary sextics and quaternary quartics in \cite{BleHauOttRanStuMR2999301}. The question which ranks can occur for extreme rays is closely related to Hilbert functions of Gorenstein ideals with socle in even degree $2d$ that contain a regular sequence in degree $d$.

Our presentation is organised as follows: In Section \ref{sec:Gorenstein}, we introduce a main tool, namely $0$-dimensional Gorenstein ideals, in a hands-on way and show that the variety of all Hankel matrices of rank at most $\binom{d+2}{2}-4$ is irreducible and has the expected codimension $10$ in the $\binom{2d+2}{2}$-dimensional space $\R[x,y,z]_{2d}^\ast$. This result is based on the Buchsbaum-Eisenbud Structure Theorem and its refined analysis by Diesel in \cite{Die}. It allows us to later identify this variety as the Zariski closure of the extreme rays of $\Sigma_{2d}^\vee$. In Section \ref{sec:ExtRays}, we conclude from the results in \cite{BlePositiveGorensteinIdeals} that $\binom{d+2}{2}-4$ is indeed the maximal rank of an extreme ray for $d\geq 4$ and construct such an extreme ray using the Cayley-Bacharach Theorem for plane curves. Section \ref{sec:Dextics} is dedicated to extreme rays of lower rank that we work out explicitly in the first intriguing case $d=5$. We also discuss the mentioned applications of the rank-stratification of the locus of extreme rays to the study of the algebraic boundary of $\Sigma_{2d}$. 

\section{Interlude on Gorenstein Ideals}\label{sec:Gorenstein}

Let us fix the following notations:
We denote by $k[\ul{x}]=k[x,y,z]$ the polynomial ring over a field $k$ generated by $3$ variables. We consider it with the standard total degree grading and denote by $k[\ul{x}]_m$ the $k$-vector space of homogeneous polynomials of degree $m$, which has dimension $\binom{m+2}{2}$.

Let $\ell\in \C[\ul{x}]_{m}^\ast$ be a linear functional on ternary forms of degree $m$. To $\ell$ and every pair of positive integers $u,v\in\N$ with $u+v=m$, we associate the bilinear form
\[
B_{\ell,u,v}\colon 
\left\{
\begin{array}[h]{rcl}
\C[\ul{x}]_u\times\C[\ul{x}]_v & \to & \C \\
(p,q) & \mapsto & \ell(pq).
\end{array}\right.
\]
The representing matrices of these bilinear forms with respect to the monomial bases are called the \emph{Catalecticant matrices} of $\ell$.

\begin{Def}\label{Def:Gorenstein}
Let $\ell\in\C[\ul{x}]_{m}^\ast$ be a linear functional. We call the homogeneous ideal $I(\ell)$ of $\C[\ul{x}]$ generated by
\[
\{p\in \C[\ul{x}]_k\colon k>m \text{ or } \ell(pq)=0 \text{ for all } q\in\C[\ul{x}]_{m-k}\}
\]
the \emph{Gorenstein ideal with socle} $\ell$. We call $m$ the \emph{socle degree} of the ideal.
\end{Def}
These ideals were studied extensively in the literature, cf.~Iarrobino-Kanev \cite{IarKanMR1735271}. Our definition is probably the most direct for $0$-dimensional Gorenstein ideals, cf.~\cite[Theorem 21.6 and Exercise 21.7]{EisenbudMR1322960}.

\begin{Rem}\label{Rem:SymHilbFunc}
The degree $u$ part of the ideal is the left-kernel of the bilinear form $B_{\ell,u,v}$ for $u\leq m$. In particular, the Hilbert function of a Gorenstein ideal $I$ with even socle degree $2d$ is symmetric around $d$, i.e.~$\hilb(I,i)=\hilb(I,2d-i)$ for all $0\leq i \leq 2d$.
\end{Rem}

We can consider the set of all Gorenstein ideals with a fixed socle degree $m$ as a projective space by identifying an ideal with its socle, which is uniquely determined by the ideal up to scaling. In this projective space, we consider the set $\gor(T)$ of all Gorenstein ideals with a given Hilbert function $T$.
\begin{Prop}
The set $\gor(T)$ of all Gorenstein ideals with socle degree $m$ and Hilbert function $T$ is a quasiprojective subvariety of the projective space of all Gorenstein ideals with socle degree $m$.
\end{Prop}

\begin{proof}
The condition to have a given Hilbert function can be expressed as rank conditions on the Catalecticant matrices, namely
\[
\rk(B_{\ell,u,v}) = T(u).
\]
\end{proof}

\begin{Rem}\label{Rem:gorTQ}
(a) The quasiprojective variety $\gor(T)$ is defined over $\Q$, because the minors of the Catalecticant matrices are polynomials with coefficients in $\Z$.\\
(b) Note that a $k$-rational point $\ell\in\gor(T)$ for a subfield $k\subset \C$ is a linear functional $\ell = \ell\otimes 1 \in k[\ul{x}]_m\otimes \C$.
\end{Rem}

\begin{Def}
We call a Hilbert function $T$ \emph{permissible} if there is a Gorenstein ideal $I\subset\C[\ul{x}]$ with Hilbert function $T$.
\end{Def}

Using the Buchsbaum-Eisenbud Structure Theorem for height $3$ Gorenstein ideals (cf.~Buchsbaum-Eisenbud \cite{BE}), Diesel proved the following.
\begin{Thm}[cf.~Diesel {\cite[Theorem 1.1 and 2.7]{Die}}]\label{Thm:DieGorT}
For every permissible Hilbert function $T$, the variety $\gor(T)$ is an irreducible unirational variety.
\end{Thm}

We will use the fact that $\gor(T)$ is unirational to determine the dimension of $\gor(T)$ for special Hilbert functions $T$. In order to do this, we need the more precise information on the unirationality of $\gor(T)$ given by Diesel. The information we need is spread out over the paper Diesel \cite{Die}. We will give a short summary with references. %Note that she works over an algebraically closed ground field, which we will follow at first. 
\begin{Rem}\label{Rem:DieselCom}
Diesel proves that for a given permissible Hilbert function $T$ there is a minimal set (with respect to inclusion) $D_{min}=(Q,P)$ of degrees of generators $Q = \{q_1,\ldots,q_u\}$ and relations $P = \{p_1,\ldots,p_u\}$ for a Gorenstein ideal with Hilbert function $T$. We assume $q_1\leq q_2\leq\ldots\leq q_u$ and $p_1\geq p_2\geq\ldots\geq p_u$. The set $\gor_{D_{min}}$ of all Gorenstein ideals with generators of degree as specified by $Q$ is a dense subset of $\gor(T)$, see the proof of \cite[Theorem 2.7 and Theorem 3.8]{Die}. Given $D_{min}$, we consider the affine space $\A^{h(E_M)}$ of skew-symmetric matrices with entries in $\C[\ul{x}]$ where the $(i,j)$-th entry is homogeneous of degree $p_j-q_i$ ($i\neq j$) and the rational map $\pi\colon \A^{h(E_M)} \ratto \gor_{D_{min}}$ that takes a matrix to the Gorenstein ideal generated by its Pfaffians. This statement uses the Buchsbaum-Eisenbud Structure Theorem, cf.~\cite{Die}, p.~367 and p.~369.
Given a Hilbert function $T$, the set $D_{min}$ of degrees of generators and relations for $T$ is determined in a combinatorial way: Given the socle degree $m$ and the minimal degree $k$ of a generator of the ideal, there is a one-to-one correspondence between permissible Hilbert functions of order $k$ and self-complementary partitions of $2k$ by $m-2k+2$ blocks, cf.~\cite[Proposition 3.9]{Die}. These partitions give the maximum number of generators, which is $2k+1$, cf.~\cite[Theorem 3.3]{Die}. To refine these sequences to $D_{min}$, we iteratively delete pairs $(q_i,q_j)$ from $Q$ and $(p_i,p_j)$ from $P$ whenever they satisfy $r_i+r_j=p_i+p_j-q_i-q_j=0$, cf.~\cite{Die}, p.~380.
\end{Rem}

We are particularly interested in Gorenstein ideals with socle in even degree $2d$ with the property that the middle Catalecticant has corank $4$, i.e.~rank $\binom{d+2}{2}-4$. The proof of the following statement is analogous to the proof of Diesel \cite[Theorem 4.4]{Die}.
\begin{Lem}\label{Lem:Corank4dim}
Let $d\geq 4$ be an integer. The projective variety $X_{-4}$ of middle Catalecticant matrices of corank at least $4$, i.e.~of rank at most $\binom{d+2}{2}-4$, is irreducible of codimension $10$ in the space of middle Catalecticant matrices. In particular, it is defined by the symmetric $(\binom{d+2}{2}-3)$-minors of the generic middle Catalecticant matrix.
\end{Lem}

\begin{proof}
Let $N = \binom{d+2}{2}$. The quasiprojective variety $S_{-4}$ of symmetric $N\times N$-matrices of rank $N-4$ has codimension $10$ in the projective space of the vector space of symmetric $N\times N$-matrices. 
Therefore the intersection $X_{-4}$ of $S_{-4}$ with the subspace of middle Catalecticant matrices has codimension at most $10$ in this linear space. We will show, that it has codimension exactly $10$ by counting dimensions of the possible $\gor(T)$ using their unirationality.

There are only two possible Hilbert functions for a Gorenstein ideal $I$ with socle degree $2d$ and $\hilb(I,d)=\binom{d+2}{2}-4$ by their symmetry, namely
\[
T_1=(1,3,6,\ldots,\binom{d+1}{2},\binom{d+2}{2}-4,\ldots),
\]
which corresponds to the case of four generators in degree $d$ and no generators of lower degree, and
\[
T_2 = (1,3,6,\ldots,\binom{d+1}{2}-1,\binom{d+2}{2}-4,\ldots),
\]
which corresponds to the case of one generator of degree $d-1$ and one generator of degree $d$. More precisely, these two Hilbert functions correspond to the self-complementary partitions of $2\times 2d$ resp.~$4\times(2d-2)$ blocks shown in Figure \ref{fig:GorensteinPartitions} by the correspondence explained in Diesel \cite[section 3.4, in paricular Proposition 3.9]{Die}.

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\filldraw[color = black!20!white] (-6,0) rectangle (-5,7);
\filldraw[color = black!20!white] (-5,0) rectangle (-4,3);
\draw[black] (-6,0) -- (-6,10) -- (-4,10) -- (-4,0) -- (-6,0);
\draw (-5,0) -- (-5,10);
\draw (-6,1) -- (-4,1);
\draw (-6,2) -- (-4,2);
\draw (-6,3) -- (-4,3);
\draw (-6,7) -- (-4,7);
\draw (-6,8) -- (-4,8);
\draw (-6,9) -- (-4,9);
\draw[very thick] (-6,10) -- (-6,7) -- (-5,7) -- (-5,3) -- (-4,3) -- (-4,0);
\draw[decorate,decoration={brace,amplitude=8pt}] (-4,7) -- (-4,3) node[midway,xshift=0.9cm] {2d-6};

\filldraw[color = black!20!white] (2,0) rectangle (3,10);
\filldraw[color = black!20!white] (3,0) rectangle (4,9);
\filldraw[color = black!20!white] (4,0) rectangle (5,1);
\draw[black] (2,0) -- (2,10) -- (6,10) -- (6,0) -- (2,0);
\draw (3,0) -- (3,10);
\draw (4,0) -- (4,10);
\draw (5,0) -- (5,10);
\draw (2,9) -- (6,9);
\draw (2,1) -- (6,1);
\draw[very thick] (3,10) -- (3,9) -- (4,9) -- (4,1) -- (5,1) -- (5,0);
\draw[decorate,decoration={brace,amplitude=8pt}] (6,9) -- (6,1) node[midway,xshift=0.9cm] {2d-4};
\end{tikzpicture}
\end{center}
\caption{The partition on the right of $2d\times 2$ blocks corresponds to $T_1$, the partition on the left of $(2d-2)\times 4$ blocks to $T_2$.}
\label{fig:GorensteinPartitions}
\end{figure}

We first consider $T_1$. The sequence of degrees of the generators for the minimal set $D_{min}$ is in this case different for $d=4$ and $d\geq 5$, namely $(4,4,4,4,6)$ for $d=4$ and $(d,d,d,d,d+1,\ldots,d+1)$ with $(2d-9)$ many generators of degree $d+1$ for $d\geq 5$, cf.~Remark \ref{Rem:DieselCom}. Since $q_i+p_i = 2d+3$, the degree matrices are
\[
\left(
\begin{array}[l]{ccccc}
0 & 3 & 3 & 3 & 1 \\
  & 0 & 3 & 3 & 1 \\
  &   & 0 & 3 & 1 \\
  &   &   & 0 & 1 \\
  &   &   &   & 0 \\
\end{array}\right),
\left(
\begin{array}[r]{cccccccc}
0 & 3 & 3 & 3 & 2 & \cdots & \cdots & 2 \\
  & 0 & 3 & 3 & \vdots &   &        & \vdots \\
  &   & 0 & 3 & \vdots &   &        & \vdots \\
  &   &   & 0 & 2 & \cdots & \cdots & 2 \\
  &   &   &   & 0 & 1      & \cdots & 1 \\
  &   &   &   &   & 0      & \ddots & \vdots \\
  &   &   &   &   &        & \ddots & 1 \\
  &   &   &   &   &        &        & 0
\end{array}\right)
\]
where the right one is of size $(2d-5)\times (2d-5)$. Every entry of the matrix can be generically chosen among the forms of the indicated degree and its Pfaffians will generate a Gorenstein ideal with Hilbert function $T_1$. Therefore, for $d=4$, we have $h(E_M)=6\dim(\C[\ul{x}]_3) + 4\dim(\C[\ul{x}]_1)=72$ and for $d\geq 5$ we have
\begin{eqnarray*}
h(E_M)& = & 6\dim(\C[\ul{x}]_3)+4(2d-9)\dim(\C[\ul{x}]_2) \\
& & + \binom{2d-9}{2}\dim(\C[\ul{x}]_1)\\
& = & 6d^2-9d-21.
\end{eqnarray*}
This is an overcount of the dimension of $\gor_{D_{min}}$ because for every choice of generators of a given ideal we get a matrix with these generators as Pfaffians. So for $d=4$, we choose a basis of a $4$-dimensional subspace of forms of degree $4$ and one generator of degree $6$ from a $\dim(\C[\ul{x}]_6) - T_1(2)=22$-dimensional space. Therefore we overcount the dimension of $\gor_{D_{min}}$ by at least $4^2+22=38$ and the dimension of $\gor(T)$ is at most $34$. Since $\dim(\P(\C[\ul{x}]_8^\ast)) = 44$, its codimension is at least $10$. For $d\geq 5$, we choose a basis of a $4$-dimensional subspace of forms of degree $d$ and $2d-9$ linearly independent generators from a space of dimension $\dim(\C[\ul{x}]_{d+1})-T_1(d-1)=2d+3$. The overcount in this case is at least $4^2+(2d-9)(2d+3)$ and the dimension of $\gor_{D_{min}}$ is at most $2d^2+3d-10$. The projective dimension of the space of middle Catalecticant matrices is $\dim(\C[\ul{x}]_{2d}^\ast)-1=2d^2+3d$, which again implies that the codimension of $\gor(T_1)$ is at least $10$. From the fact that it can be at most $10$, it follows that it is exactly $10$.

We now repeat the count for the Hilbert function $T_2$. In this case, $D_{min} = \{Q_{min},P_{min}\} = \{(d-1,d,d+1,d+1,\ldots,d+1),(d+4,d+3,d+2,d+2,\ldots,d+2)\}$ with $(2d-5)$ times the entry $d+1$ in $Q_{min}$ and $d+2$ in $P_{min}$, cf.~Figure \ref{fig:GorensteinPartitions}. Therefore, the degree matrix is
\[
\left(
\begin{array}[h]{cccccc}
0 & 4 & 3 & \cdots & \cdots & 3 \\
  & 0 & 2 & \cdots & \cdots & 2 \\
  &   & 0 & 1      & \cdots & 1 \\
  &   &   & 0      & \ddots & \vdots \\
  &   &   &        &        & 1 \\
  &   &   &        &        & 0
\end{array}\right)
\]
which is of size $(2d-3)\times (2d-3)$. We compute
\begin{eqnarray*}
h(E_M) & = & \dim(\C[\ul{x}]_4)+(2d-5)\dim(\C[\ul{x}]_3)+(2d-5)\dim(\C[\ul{x}]_2) \\
& & +\binom{2d-5}{2}\dim(\C[\ul{x}]_1) \\
& = & 6d^2-d-20.
\end{eqnarray*}
 Here we choose one generator of degree $d-1$, one generator of degree $d$ from a $4$-dimensional space and $(2d-5)$ generators from a $\dim(\C[\ul{x}]_{d+1})-T_2(d-1)=(2d+4)$-dimensional space. Therefore the dimension of $\gor(T_2)$ is at most $6d^2-d-20-4-(2d-5)(2d+4)=2d^2+d-4$. The codimension is at least $2d+4\geq 12$.
So $\gor(T_2)$ cannot be an irreducible component of $X_{-4}$ and we conclude that $\gor(T_2)\subset \cl(\gor(T_1))$.
In summary, $\gor(T_1)$ is a dense subset of $X_{-4}$ and $X_{-4}$ is irreducible (cf.~Diesel \cite[Theorem 2.7]{Die}) and has the expected codimension $10$ in the space of middle Catalecticant matrices.
\end{proof}

The tangent space to the quasiprojective variety $\gor(T)$ for a permissible Hilbert function $T$ at a Gorenstein ideal $I$ can be described in terms of the ideal. We identify $\C[\ul{x}]_m$ with its dual space by using the apolar bilinear form, i.e.~we identify a monomial $x^\alpha\in\C[\ul{x}]_m$ with the linear form $p\mapsto \frac{1}{\alpha!} \frac {\partial^{|\alpha|}}{\partial x^\alpha}p$ that takes a polynomial $p=\sum p_\beta x^\beta$ to $p_\alpha$. Using this identification, we can state a characterisation of the tangent space to $\gor(T)$ at an ideal $I$ in terms of this ideal.
\begin{Thm}[Iarrobino-Kanev {\cite[Theorem 3.9 and 4.21]{IarKanMR1735271}}]
\label{Thm:GorTTangentSpace}
Let $T$ be a permissible Hilbert function. The quasiprojective variety $\gor(T)$ is smooth.
Let $\ell\in\C[\ul{x}]_m^\ast$ be a linear functional such that the corresponding Gorenstein ideal $I=I(\ell)$ has Hilbert function $T$. Then the tangent space to $\gor(T)$ at $\ell$ is
\[
( (I^2)_m)^\perp\subset \C[\ul{x}]_m.
\]
\end{Thm}

The irreducible variety $X_{-4}$ of middle Catalecticant matrices of corank at least $4$ is defined over $\Q$ and we will later show that it has a smooth rational point, i.e.~a point with rational coordinates. Therefore, the real points of $X_{-4}$ are Zariski-dense in it and the above statement of Theorem \ref{Thm:GorTTangentSpace} also applies to real points of $X_{-4}$, cf.~\cite[Section 2.8]{BochnakMR1659509}.

\section{Extreme Rays of Maximal Rank and Positive Gorenstein Ideals}\label{sec:ExtRays}
In this section, we recapitulate bounds on the rank of Hankel matrices of extreme rays of $\Sigma_{2d}^\vee$ which are not point evaluations. The lower bound and its tightness are proved in Blekherman \cite[Theorem 2.1]{BlePositiveGorensteinIdeals}. We constructively establish tightness of the upper bound. We show that the Zariski closure of the set of extreme rays is the variety of Hankel matrices of corank at least $4$, which is irreducible; in particular, it is (at least set-theoretically) defined by the symmetric $r\times r$ minors of the generic Hankel matrix, where $r=\binom{d+2}{2}-3$.

To a linear functional $\ell \in \R[\ul{x}]_m^\ast$, we associate the bilinear form 
\[
B_{\ell}\colon 
\left\{
\begin{array}[h]{rcl}
\R[\ul{x}]_d\times\R[\ul{x}]_d & \to & \R \\
(p,q) & \mapsto & \ell(pq),
\end{array}\right.
\]
whose representing matrix with respect to the monomial bases is called the \emph{Hankel matrix} of $\ell$.

One of the main results of Blekherman is a characterisation of extreme rays of $\Sigma_{2d}^\vee$ by the associated Gorenstein ideals.
\begin{Prop}[Blekherman {\cite[Lemma 2.2]{BleNP}} and {\cite[Proposition 4.2]{BlePositiveGorensteinIdeals}}]
\label{Prop:ExtremeRays}
(a) A linear functional $\ell\in\R[\ul{x}]_{2d}^\ast$ spans an extreme ray of $\Sigma_{2d}^\vee$ if and only if the bilinear form $B_\ell$ is positive semi-definite and the degree $d$ part $I(\ell)_d$ of the Gorenstein ideal $I(\ell)$ is maximal with respect to inclusion over all Gorenstein ideals with socle degree $2d$.\\
(b) Let $I$ be a Gorenstein ideal with socle degree $2d$. Then $I_d$ is maximal with respect to inclusion over all Gorenstein ideals with socle degree $2d$ if and only if the degree $2d$ part of the ideal generated by $I_d$ is a hyperplane in $\R[\ul{x}]_{2d}$. In this case, it is equal to $I_{2d}$.
\end{Prop}

Lower bounds on the ranks for extreme rays were established by Blekherman.
\begin{Thm}[Blekherman {\cite[Theorem 2.1]{BlePositiveGorensteinIdeals}}]\label{Thm:ExtremeRays}
Let $d\geq 3$ and $\ell\in \Sigma_{2d}^\vee$ and suppose $\R_+\ell$ is an extreme ray. Then the rank $r$ of $B_\ell$ is $1$, in which case $\ell$ is a point evaluation, or its rank is at least $3d-2$.
These bounds are tight and extreme rays $\Sigma_{2d}^\vee$ of rank $3d-2$ can be explicitly constructed.
\end{Thm}

From Blekherman's work, we can easily deduce an upper bound.
\begin{Thm}\label{Thm:MaxRank}
Let $\ell\in\Sigma_{2d}^\vee$, $d\geq 4$ and suppose $\R_+\ell$ is an extreme ray. The rank of $B_\ell$ is at most $\binom{d+2}{2}-4$, i.e.~the corank is at least $4$.
\end{Thm}

\begin{proof}
Since $\R_+\ell$ is an extreme ray, we know that the degree $2d$ part of the ideal generated by $I(\ell)_d$ is a hyperplane in the space of forms of degree $2d$.
The dimension of the space $\R[\ul{x}]_d I(\ell)_d$ is bounded by $\dim(\R[\ul{x}]_d) \dim(I(\ell)_d)=\binom{d+2}{2}\crk(B_\ell)$. In case $\crk(B_\ell)\leq 3$ and $d\geq 5$, this bound is smaller than the dimension $\binom{2d+2}{2}-1$ of a hyperplane in $\R[\ul{x}]_{2d}$. The case $\crk(B_\ell) \leq 3$ and $d=4$ needs a more precise count: Suppose that $\crk(B_\ell)=3$ and the kernel of $B_\ell$ is generated by $f_1,f_2,f_3$. Then the dimension of the space $\R[\ul{x}]_4 I(\ell)_4$ is bounded by $3\dim(\R[\ul{x}]_4)-3 = 42<45-1 = \dim(\R[\ul{x}]_8) -1$ because there are the $3$ obvious relations, namely $f_if_j-f_jf_i = 0$ for $i\neq j$. 
\end{proof}

\begin{Rem}
The upper bound in the case $d=3$ is corank $3$, which agrees with the lower bound.
\end{Rem}

A main tool in this section is the Cayley-Bacharach Theorem.
\begin{Thm}[Cayley-Bacharach, cf.~Eisenbud-Green-Harris {\cite[CB5]{EisGreHar}}]
Let $X_1,X_2\subset\P^2$ be plane curves defined over $\R$ of degree $d$ and $e$ intersecting in $d\cdot e$ points. Set $s= d+e-3$ and decompose $X_1\cap X_2 = \Gamma_1\cup \Gamma_2$ into two disjoint sets defined over $\R$. Then for all $k\leq s$, the following equality holds
\begin{eqnarray*}
& \dim(\I(\Gamma_1)_k) - \dim(\I(X_1\cap X_2)_k) = \\
& |\Gamma_2| - \dim \lspan \{ {\rm Re}\ev_x, {\rm Im}\ev_x \in\R[\ul{x}]_{s-k}^\ast\colon x\in\Gamma_2\}.
\end{eqnarray*}
The left hand side is the dimension of the space of forms of degree $k$ vanishing on $\Gamma_1$ modulo the subspace of forms vanishing in every point of $X_1\cap X_2$. The right hand side is the linear defect of point evaluations on forms of dual degree $s-k$ at points of $\Gamma_2$.
\end{Thm}

Probably the most famous instance of this theorem is the following application to the complete intersection of two cubic curves, stated here for a totally real intersection.
\begin{Exm}
Suppose $X_1,X_2\subset\P^2$ are plane cubic curves intersecting in $9$ points. Then $d=e=3$ and so $s=3$. Pick $\Gamma_2 = \{P\}$ for any intersection point $P$ and put $\Gamma_1 = (X_1\cap X_2) \setminus \{P\}$. Let us consider $k=3$ and compute the right hand side of the Cayley-Bacharach equality: Since $\dim \lspan\{\ev_x\in\R[\ul{x}]_0^\ast\colon x\in\Gamma_2\} = 1$, we conclude
\[
\dim(\I(\Gamma_1)_3) - \dim(\I(X_1\cap X_2)_3) = 0,
\]
which means that every cubic form that vanishes in the $8$ points of $\Gamma_1$ also vanishes at the ninth point $P$ of the intersection. In other words, the point evaluation $\ev_P\in\R[\ul{x}]_3^\ast$ lies in the subspace $U_{\Gamma_1}$ spanned by the eight point evaluations $\{\ev_x\in\R[\ul{x}]_3^\ast\colon x\in\Gamma_1\}$. The annihilator of $U_{\Gamma_1}$ is the $2$-dimensional subspace of $\R[\ul{x}]_3$ spanned by the defining equations of $X_1$ and $X_2$. Since this is true for any point $P\in X_1\cap X_2$, we conclude, that there is a unique linear relation among the point evaluations $\{\ev_x\in\R[\ul{x}]_3^\ast\colon x\in X_1\cap X_2\}$ and all coefficients of this relation are non-zero.
\end{Exm}

Using the Cayley-Bacharach Theorem, we will first show that there are extreme rays of corank $4$ under the following constraint on the degree. 
\begin{Const}\label{Const:ellipse}
Let $d\geq 4$. There is a unique conic $C$ going through the following six points in the plane: $(0,0),(1,0),(0,1),(d-1,d-1),(d-2,d-1),(d-1,d-2)$; its equation is given by $C = \V(x^2+y^2-\frac{2(d-2)}{d-1}xy-x-y)$. From now on, we assume that this conic does not go through any other integer point. The only exceptional cases in the interval $\{4,5,\ldots,100\}$ are: $9,19,21,29,33,34,36,40,49,51,57,61,73,78,79,81,89,99$.
\end{Const}

\begin{Prop}\label{Prop:CB}
Set $L_1=\prod_{j=0}^{d-1}(x-jz)$ and $L_2=\prod_{j=0}^{d-1}(y-jz)$ and let $\Gamma=\V(L_1,L_2)=\{(j:k:1)\colon j,k=0,\ldots,d-1\}$ be the intersection of their zero sets in $\P^2$. Split these points into
\[
\Gamma_2 = \{(x:y:1)\colon x+y=2\}\cup\bigcup_{j=1}^{d-4}\{(x:y:1)\colon x+y=d+j\}
\]
and $\Gamma_1=\Gamma\setminus\Gamma_2$.
Then there is a unique linear relation $\sum_{v\in\Gamma_1} u_v{\rm ev}_v=0$ among the point evaluations on forms of degree $d$ at points of $\Gamma_1$ and all coefficients $u_v\in\R$ in this relation are non-zero.
The set of all forms of degree $d$ vanishing on $\Gamma_1$ is a $3$-dimenional space spanned by $L_1,L_2$ and a form $p$ which is non-zero at any point of $\Gamma_2$.
\end{Prop}
See Figure \ref{fig:ExtRayC4} for the case $d=5$ and Figure \ref{fig:perturbation} for the case $d=9$.

\begin{proof}
First observe that there is a unique (up to scaling) form of degree $d-3$ vanishing on $\Gamma_2$, namely $(x+y-2z)\prod_{j=1}^{d-4}(x+y-(d+j)z)$, the product of diagonals defining $\Gamma_2$: Indeed, suppose $f$ is a form of degree $d-3$ vanishing on $\Gamma_2$, then it intersects the line $x+y=d+1$ in $d-2$ integer points. Therefore it vanishes identically on it and we can divide $f$ by this linear polynomial and get a form of degree $d-4$ vanishing on $d-3$ points on the line $x+y=d+2$. Inductively, we conclude that $f$ is (again up to scaling) the claimed product of linear forms. Therefore, by the Cayley-Bacharach Theorem, the space of forms of degree $d$ vanishing on $\Gamma_1$ is $3$-dimensional,
so it is spanned by $L_1, L_2$ and a third form $p$. We will explicitly construct this form: Let $p$ be the product of the linear forms $x+y-jz$ for $j=3,\ldots,d$ and of the ellipse $\V(x^2+y^2-\frac{2(d-2)}{(d-1)}xy-x-y)$ passing through the six points $(0,0),(1,0),(0,1),(d-2,d-1),(d-1,d-1)$ and $(d-1,d-2)$. By construction, $p$ vanishes on $\Gamma_1$, is of degree $d$ and does not vanish on all of $\Gamma$. Therefore $\{L_1,L_2,p\}$ is a basis of the space of forms of degree $d$ vanishing on $\Gamma_1$. By assumption on $d$, the form $p$ does not vanish on any point of $\Gamma$ other than the six mentioned above. 

Note that $|\Gamma_1| = \binom{d+2}{2}-2$, because $|\Gamma_1| = d^2 - |\Gamma_2| = d^2 - (3+\sum_{j=1}^{d-4}(d-1-j))=d^2-\binom{d-1}{2}$.
So the fact that the space of forms of degree $d$ vanishing on $\Gamma_1$ is $3$-dimensional implies that there is a unique linear relation among the point evalutaions on forms of degree $d$ at points of $\Gamma_1$. To see that all coefficients $u_v$ in the relation $\sum_{v\in\Gamma_1}u_v{\rm ev}_v=0$ are non-zero, note that the unique form $f$ of degree $d-3$ vanishing on $\Gamma_2$ does not vanish on any point of $\Gamma_1$. Therefore, there is no form of degree $d-3$ vanishing on $\Gamma_2\cup\{v_0\}$ for any $v_0\in\Gamma_1$ and Cayley-Bachrach implies that the point evaluations $\{ {\rm ev}_v\colon v\in \Gamma_1\}\setminus\{ {\rm ev}_{v_0}\}$ are linearly independent.
\end{proof}

\begin{Lem}\label{Lem:ExtremeRaysC4}
There is an extreme ray $\R_+\ell$ of $\Sigma_{2d}^\vee$ such that $B_\ell$ has corank $4$. The Hilbert function of the ideal $I(\ell)$ is $\hilb(I(\ell),j)=\binom{j+2}{2} = \hilb(I(\ell),2d-j)$ for $0\leq j<d$ and $\hilb(I(\ell),d)=\binom{d+2}{2}-4$.
\end{Lem}

\begin{proof}
Let $L_1,L_2,p$ be as in Proposition \ref{Prop:CB} and consider the splitting $\V(L_1,L_2) = \Gamma =  \Gamma_1 \cup \Gamma_2$ of the points defined there. Pick a point $P\in\Gamma_1$ and set $\Lambda = \Gamma_1\setminus\{P\}$. We claim that the linear functional
\[
\ell = \sum_{v\in \Lambda} \ev_v - \frac{u_P^2}{\sum_{v\in\Lambda}u_v^2} \ev_P,
\]
where $u_v$ are the coefficients of the Cayley-Bacharach relation as in Proposition \ref{Prop:CB}, is an extreme ray of $\Sigma_{2d}^\vee$ and that the corresponding Hankel matrix $B_\ell$ has corank $4$.

First note that $B_\ell$ is positive semi-definite because
\begin{eqnarray*}
\ell(f^2) & = & \sum_{v\in\Lambda} f(v)^2 - \frac{u_P^2}{\sum_{v\in\Lambda} u_v^2} f(P)^2 \\
& = & \sum_{v\in\Lambda} f(v)^2 - \frac{u_P^2}{\sum_{v\in\Lambda} u_v^2} \frac{1}{u_P^2} \left(\sum_{v\in\Lambda} u_vf(v)\right)^2 \\
& = & \|(f(v)_{v\in\Lambda}\|^2 - \left| \left\langle \frac{1}{\|(u_v)_{v\in\Lambda}\|} (u_v)_{v\in\Lambda} , (f(v))_{v\in\Lambda} \right\rangle \right|^2 \\
& \geq & 0
\end{eqnarray*}
by the Cauchy-Schwarz inequality for all polynomials $f\in\R[\ul{x}]_d$. More precisely, $\ell(f^2)$ is zero for a form $f$ not identically zero on $\Gamma$ if and only if $f(v)=\alpha u_v$ for all $v\in\Lambda$ and some $\alpha\in\R^\ast$. Therefore, the degeneration space of the Hankel matrix is spanned by $L_1,L_2,p$ and the form uniquely determined (modulo $L_1,L_2,p$) by $f(v)=u_v$ for all $v\in\Lambda$; it has dimension $4$ as desired. Indeed, the form $f$ is uniquely determined because $\{\ev_x\in (\R[\ul{x}]_d/\lspan(L_1,L_2,p))^\ast\colon x\in\Lambda\}$ is a basis.

We now prove extremality of $\ell$ in $\Sigma_{2d}^\vee$ by checking the characterisation that $I(\ell)_d$ generates a hyperplane in the vector space of forms of degree $2d$, cf.~Blekherman \cite[Proposition 4.2]{BlePositiveGorensteinIdeals}. As a first step, we show that $\langle L_1,L_2,p \rangle_{2d-3}$ has codimension $|\V(L_1,L_2,p)|=|\Gamma_1|$. So suppose $a_1 L_1+a_2 L_2 + bp=0$, where $a_1,a_2,b\in\R[\ul{x}]_{d-3}$ are forms of degree $d-3$. By evaluating at points of $\Gamma_2$, we conclude that $b$ is the uniquely determined form of degree $d-3$ vanishing on $\Gamma_2$, cf.~proof of Proposition \ref{Prop:CB}. Since $L_1$ and $L_2$ are coprime, this is a unique syzygy and we conclude
\[
\dim(\langle L_1,L_2,p\rangle_{2d-3}) = 3\binom{d-1}{2} -1 = \frac32(d^2-3d+2)-1,
\]
which means codimension $|\Gamma_1|=d^2-\binom{d-1}{2}$ in $\R[\ul{x}]_{2d-3}$.
In particular, the codimension of $\langle L_1,L_2,p\rangle_{2d}$ in $\R[\ul{x}]_{2d}$ is also $|\Gamma_1|$ because the point evaluations $\{\ev_v\colon v\in\Gamma_1\}$ are linearly independent on forms of degree $2d-3$ and consequently also on forms of degree $2d$.

Now suppose $a_1L_1 + a_2L_2+bp + cf=0$ for forms $a_1,a_2,b,c\in\R[\ul{x}]_d$ of degree $d$. Evaluation at points of $\Gamma_1$ implies that $c$ lies in the span of $L_1,L_2,p$. So we have three syzygies and the codimension of $\langle L_1,L_2,p,f\rangle_{2d}$ is $|\Gamma_1| - \binom{d+2}{2}+3=1$, as desired.
\end{proof}

\begin{Exm}\label{Exm:Hankel}
We follow the construction in Proposition \ref{Prop:CB} and Lemma \ref{Lem:ExtremeRaysC4} in the case $d=5$. Then $\Gamma=\V(L_1,L_2)$ consists of the $25$ points $(i:j:1)\in\P^2$ where $i,j=0,\ldots,4$, see Figure \ref{fig:ExtRayC4}. The six points on the two lines $x+y=2$ and $x+y=6$ are the points of $\Gamma_2$. Indeed, the point evaluations at the $19$ points of $\Gamma_1 = \Gamma\setminus \Gamma_2$ on forms of degree $5$ satisfy a unique linear relation, namely


$\begin{pmatrix}
-1 &   3 &   0 &  -5 &  3 \\
 3 & -16 &  18 &   0 & -5 \\
 0 &  18 & -36 &  18 &  0 \\
-5 &   0 &  18 & -16 &  3 \\
 3 &  -5 &   0 &   3 & -1 \\
\end{pmatrix}$,

where the $(i,j)$-th entry of this matrix is the coefficient of the point evaluation at $(5-i:j-1:1)$ in the linear relation, i.e.~visually, it is the coefficient corresponding to the points in the $5\times 5$-grid seen in Figure \ref{fig:ExtRayC4}.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\filldraw (0,0) circle(2pt);
\filldraw (0,1) circle(2pt);
\filldraw (0,2) circle(2pt);
\filldraw (0,3) circle(2pt);
\filldraw (0,4) circle(2pt);
\filldraw (1,0) circle(2pt);
\filldraw (1,1) circle(2pt);
\filldraw (1,2) circle(2pt);
\filldraw (1,3) circle(2pt);
\filldraw (1,4) circle(2pt);
\filldraw (2,0) circle(2pt);
\filldraw (2,1) circle(2pt);
\filldraw (2,2) circle(2pt);
\filldraw (2,3) circle(2pt);
\filldraw (2,4) circle(2pt);
\filldraw (3,0) circle(2pt);
\filldraw (3,1) circle(2pt);
\filldraw (3,2) circle(2pt);
\filldraw (3,3) circle(2pt);
\filldraw (3,4) circle(2pt);
\filldraw (4,0) circle(2pt);
\filldraw (4,1) circle(2pt);
\filldraw (4,2) circle(2pt);
\filldraw (4,3) circle(2pt);
\filldraw (4,4) circle(2pt);
\draw (-1,3) -- (3,-1);
\draw (1,5) -- (5,1);
\end{tikzpicture}
\end{center}
\caption{The construction of an extreme ray of $\Sigma_{2d}^\vee$ of corank $4$ for $d=5$.}
\label{fig:ExtRayC4}
\end{figure}
The $21\times 21$ Hankel matrix can be exactly computed using a computer algebra system. In Mathematica, the following code will do the job:

\begin{verbatim}
d = 5;
m = MonomialList[(x+y+z)^d];
q1 = x (x - z) (x - 2 z) (x - 3 z) (x - 4 z);
q2 = y (y - z) (y - 2 z) (y - 3 z) (y - 4 z);
Pevalall = Solve[{q1 ==  0, q2 ==  0, z ==  1}, {x, y, z}];
Pevalfoo = Select[Pevalall, ({y + x - 2 z} /. #) != {0} &];
Peval = Select[Pevalfoo, ({y + x - 6 z} /. #) != {0} &];
Peval0 = Drop[Peval, -1];

H = Transpose[{m}].{m};
P = Peval0;
Q = Sum[H /. P[[i]], {i, 1, Length[P]}];
Qp = H /. Peval[[-1]];
l = Norm[CB]^2;
Hankel = Q - 1/l (rel[[1]][[-1]])^2 Qp;
\end{verbatim}
We set up the monomial basis \texttt{m} and the totally real complete intersection of $25$ points, where $\texttt{q1}=L_1$ and $\texttt{q2} = L_2$. The two lines using the \texttt{Select}-command remove the points on the two diagonals $x+y=2$ and $x+y=6$, so $\Gamma_1 = \texttt{Peval}$. With the \texttt{Drop}-command, we remove one of the points from the list. 
The matrix \texttt{H} is the general Hankel matrix and $Q$ is the Hankel matrix of the linear functional $\sum_{v\in\Gamma_1\setminus \texttt{Peval[[-1]]}} \ev_v$ and \texttt{Qp} the Hankel matrix of the point evaluation at \texttt{Peval[[-1]]}. So \texttt{Hankel} is the Hankel matrix of the extreme ray that we constructed.
\end{Exm}

\begin{Rem}
Note that the proof of the Lemma \ref{Lem:ExtremeRaysC4} shows that the face of the cone $\Sigma_{2d}$ of sums of squares exposed by the constructed extreme ray consists of the sums of squares of polynomials in $I(\ell)_d$.
\end{Rem}

The fact that the conic vanishes in additional integer points on the $d\times d$ grid defined by the products of linear forms $L_1$ and $L_2$ in Proposition \ref{Prop:CB} destroys the extremality of the constructed linear functional because we get additional syzygies among the generators of the corresponding Gorenstein ideal. In order to deal with this problem, we will make a perturbation to our point arrangement. First, we want to observe the following fact, which motivates why we should be able to get around this obsatcle by perturbation:
\begin{Rem}\label{Rem:CBApplication}
Consider the setup in Proposition \ref{Prop:CB} and suppose the conic $C$ vanishes in additional integer points in the $d\times d$ integer grid $\Gamma=\V(L_1)\cap \V(L_2)$. Pick such a point $P\in\Gamma$. Then every form of degree $d$ vanishing on $\Gamma_1$ will also vanish at $P$ because $L_1$, $L_2$ and the third form $p$, which is the product of lines and the conic, form a basis of this space. By the Theorem of Cayley-Bacharach applied to $\Gamma = \Gamma_1'\cup\Gamma_2'$ for $\Gamma_1' = \Gamma_1 \cup\{P\}$ and $\Gamma_2' = \Gamma_2\setminus\{P\}$, there is a unique linear relation among the point evaluations at points of $\Gamma_2'$ on forms of degree $d-3$. In particular, the coefficient of the point evaluation at $P$ in the unique linear relation among point evaluations at $\Gamma_2$ on forms of degree $d-3$ is zero. The converse is also true by Cayley-Bacharach, so we have:\\
The conic $C$ vanishes in a point in $P\in\Gamma_2$ if and only if the coefficient of the point evaluation at $P$ in the unique linear relation among $\{\ev_v\in \R[\ul{x}]_{d-3}^\ast \colon v\in\Gamma_2\}$ is zero. This seems to be a non-generic property and we will indeed show that we can make all coefficients in the linear relation among these point evaluations non-zero by a careful perturbation of $L_1$ and $L_2$.
\end{Rem}

We now drop the assumptions on $d$ made in \ref{Const:ellipse} and prove Lemma \ref{Lem:ExtremeRaysC4} for all $d\geq 4$:
\begin{Lem}\label{Lem:ExRayC4Per}
For any $d\geq 4$, there is an extreme ray $\R_+\ell$ of $\Sigma_{2d}^\vee$ such that $B_\ell$ has corank $4$. The Hilbert function of the ideal $I(\ell)$ is $\hilb(I(\ell),j) = \binom{j+2}{2}$ for $0\leq j<d$ and $\hilb(I(\ell),d) = \binom{d+2}{2}-4$.
\end{Lem}

\begin{proof}
We start as above with the products of linear forms $L_1 =\prod_{j=0}^{d-1}(x-jz)$ and $L_2 =\prod_{j=0}^{d-1}(y-jz) $ and denote by $\Gamma$ the complete intersection $\V(L_1)\cap \V(L_2)$. Split $\Gamma$ into
\[
\Gamma_2 = \{(x:y:1)\colon x+y=2\}\cup\bigcup_{j=1}^{d-4}\{(x:y:1)\colon x+y=d+j\}
\]
and $\Gamma_1=\Gamma\setminus\Gamma_2$. Then the space of forms of degree $d$ vanishing on $\Gamma_1$ has dimension $3$. Let $p$ be the uniquely determined form of degree $d$ such that $L_1,L_2,p$ is a basis of this space. By Cayley-Bacharach, we know that there is a unique relation among the point evaluations $\{\ev_x\in \R[\ul{x}]_{d-3}^\ast \colon x\in\Gamma_2\}$, say
\[
\sum_{x\in\Gamma_2} w_x\ev_x=0.
\]
Note that by the preceding Remark \ref{Rem:CBApplication}, the coefficient of $\ev_{(1:1:1)}$ is non-zero.
Set $\Gamma_1'=\Gamma_1\cup \{(1:1:1)\}$ and $\Gamma_2'=\Gamma_2'\setminus\{(1:1:1)\}$. Then the point evaluations $\{\ev_x\in\R[\ul{x}]_{d-3}^\ast\colon x\in\Gamma_2'\}$ are linearly independent and span a hyperplane $H$ in $\R[\ul{x}]_{d-3}^\ast$. So there is a unique form $q$ of degree $d-3$ vanishing on $\Gamma_2'$, namely the one vanishing on all of $\Gamma_2$, i.e.~$q=(x+y-2z)\prod_{j=1}^{d-4}(x+y-(d+j)z)$.

We will now perturb the point $(1:1:1)$ along the line $x+y=2$, see Figure \ref{fig:perturbation} for a visualisation in case $d=9$: Let $v_t := (t,2-t)$. Of course, $q(v_t)=0$ for every $t\in\R$, i.e.~the point evaluation $\ev_{v_t}\in\R[\ul{x}]_{d-3}^\ast$ lies in the hyperplane spanned by the point evaluations at $\Gamma_2'$; write
\[
\ev_{v_t} = \sum_{x\in\Gamma_2'} \alpha_x(t)\ev_x,
\]
where the coefficients $\alpha_x(t)$ are rational functions of the parameter $t$.

Suppose there is a point $P\in\Gamma_2'$ such that $\alpha_P(t)=0$ for all $t\in\R$. Then $\ev_{v_t}\in\lspan(\ev_v\colon v\in\Gamma_2'\setminus\{P\})$. Dually this means, that there is a form $f_P$ of degree $d-3$, uniquely determined modulo $q$, such that
$f_P(P)=1$,
$f_P(v)=0$ for all $v\in\Gamma_2'\setminus\{P\}$ and consequently
$f_P(v_t) = 0$ for all $t\in\R$.
Such a form cannot exist: Since $v_t$ ranges over the whole line defined by $x+y=2$, the form $f_P$ vanishes identically on this line; so we can factor it out. Furthermore, $f_P$ vanishes identically on every diagonal defining $\Gamma_2$ to the left of $P$, i.e.~$f_P(x,j-x)=0$ for all $d<j<P_1+P_2$ because it has too many zeros on these lines from $\Gamma_2'$.

Now $\Gamma_2'\cap\{x+y=P_1+P_2\}$ consists of $2d-1-P_1-P_2$ many points. We have already established $P_1+P_2-d$ linear factors of $f_P$, so the remaining cofactor has degree $2d-P_1-P_2-3$. Therefore, $f_P$ vanishes identically on this line, which is a contradiction because it contains $P$.

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\filldraw (0,0) circle(2pt); \filldraw (0,2) circle(2pt); \filldraw (0,3) circle(2pt); \filldraw (0,4) circle(2pt); \filldraw (0,5) circle(2pt); \filldraw (0,6) circle(2pt); \filldraw (0,7) circle(2pt); \filldraw (0,8) circle(2pt); \filldraw (2,0) circle(2pt); \filldraw (2,2) circle(2pt); \filldraw (2,3) circle(2pt); \filldraw[red] (2,4) circle(2pt); \filldraw (2,5) circle(2pt); \filldraw (2,6) circle(2pt); \filldraw (2,7) circle(2pt); \filldraw (2,8) circle(2pt); \filldraw (3,0) circle(2pt); \filldraw (3,2) circle(2pt); \filldraw (3,3) circle(2pt); \filldraw (3,4) circle(2pt); \filldraw (3,5) circle(2pt); \filldraw (3,6) circle(2pt); \filldraw (3,7) circle(2pt); \filldraw (3,8) circle(2pt); \filldraw (4,0) circle(2pt); \filldraw[red] (4,2) circle(2pt); \filldraw (4,3) circle(2pt); \filldraw (4,4) circle(2pt); \filldraw (4,5) circle(2pt); \filldraw[red] (4,6) circle(2pt); \filldraw (4,7) circle(2pt); \filldraw (4,8) circle(2pt); \filldraw (5,0) circle(2pt); \filldraw (5,2) circle(2pt); \filldraw (5,3) circle(2pt); \filldraw (5,4) circle(2pt); \filldraw (5,5) circle(2pt); \filldraw (5,6) circle(2pt); \filldraw (5,7) circle(2pt); \filldraw (5,8) circle(2pt); \filldraw (6,0) circle(2pt); \filldraw (6,2) circle(2pt); \filldraw (6,3) circle(2pt); \filldraw[red] (6,4) circle(2pt); \filldraw (6,5) circle(2pt); \filldraw (6,6) circle(2pt); \filldraw (6,7) circle(2pt); \filldraw (6,8) circle(2pt); \filldraw (7,0) circle(2pt); \filldraw (7,2) circle(2pt); \filldraw (7,3) circle(2pt); \filldraw (7,4) circle(2pt); \filldraw (7,5) circle(2pt); \filldraw (7,6) circle(2pt); \filldraw (7,7) circle(2pt); \filldraw (7,8) circle(2pt); \filldraw (8,0) circle(2pt); \filldraw (8,2) circle(2pt); \filldraw (8,3) circle(2pt); \filldraw (8,4) circle(2pt); \filldraw (8,5) circle(2pt); \filldraw (8,6) circle(2pt); \filldraw (8,7) circle(2pt); \filldraw (8,8) circle(2pt);
\filldraw (3/4,0) circle(2pt); \filldraw (3/4,5/4) circle(2pt); \filldraw (3/4,2) circle(2pt); \filldraw (3/4,3) circle(2pt); \filldraw (3/4,4) circle(2pt); \filldraw (3/4,5) circle(2pt); \filldraw (3/4,6) circle(2pt); \filldraw (3/4,7) circle(2pt); \filldraw (3/4,8) circle(2pt);
\filldraw (0,5/4) circle(2pt); \filldraw (2,5/4) circle(2pt); \filldraw (3,5/4) circle(2pt); \filldraw (4,5/4) circle(2pt); \filldraw (5,5/4) circle(2pt); \filldraw (6,5/4) circle(2pt); \filldraw (7,5/4) circle(2pt); \filldraw (8,5/4) circle(2pt);
\filldraw[black!30!white] (0,1) circle(2pt); \filldraw[black!30!white] (1,1) circle(2pt); \filldraw[black!30!white] (2,1) circle(2pt); \filldraw[black!30!white] (3,1) circle(2pt); \filldraw[black!30!white] (4,1) circle(2pt); \filldraw[black!30!white] (5,1) circle(2pt); \filldraw[black!30!white] (6,1) circle(2pt); \filldraw[black!30!white] (7,1) circle(2pt); \filldraw[black!30!white] (8,1) circle(2pt);
\filldraw[black!30!white] (1,0) circle(2pt); \filldraw[black!30!white] (1,2) circle(2pt); \filldraw[black!30!white] (1,3) circle(2pt); \filldraw[black!30!white] (1,4) circle(2pt); \filldraw[black!30!white] (1,5) circle(2pt); \filldraw[black!30!white] (1,6) circle(2pt); \filldraw[black!30!white] (1,7) circle(2pt); \filldraw[black!30!white] (1,8) circle(2pt);
\draw (-1,3) -- (3,-1);
\draw (1,9) -- (9,1);
\draw (2,9) -- (9,2);
\draw (3,9) -- (9,3);
\draw (4,9) -- (9,4);
\draw (5,9) -- (9,5);
\draw[black!30!white] (1,-0.2) -- (1,8.2);
\draw[black!30!white] (-0.2,1) -- (8.2,1);
\draw[rotate=45,black!30!white] (5.65685,0) ellipse (5.65685cm and 1.46059cm);
\end{tikzpicture}
\end{center}
\caption{A picture of the perturbation for general $d\geq 4$ shown for the first critical case $d=9$: The black points and the four red points are the perturbed point configuration for which our construction works. The four red points are the additional points through which the grey ellipse goes.}
\label{fig:perturbation}
\end{figure}

So there is an $\epsilon>0$ such that for all $t\in (1-\epsilon,1)$, all coefficients of the linear relation
\[
\ev_{v_t} = \sum_{v\in\Gamma_2'}\alpha_v(t)\ev_v
\]
are non-zero. Pick a $t_0$ in this interval and consider the totally real complete intersection $\Gamma=\V(L_1')\cap\V(L_2')$ for $L_1 = x(x-t_0z)\prod_{j=2}^{d-1}(x-jz)$ and $L_2 = y(y-(2-t_0)z)\prod_{j=2}^{d-1}(x-jz)$ and argue as above: we split the points into $\Gamma_1$ and $\Gamma_2$, where $\Gamma_2$ is the same union of diagonals as above. The Theorem of Cayley-Bacharach then implies the existence of a form of degree $d$ vanishing on $\Gamma_1$ and not identically on $\Gamma$. In fact, by Remark \ref{Rem:CBApplication}, this form does not vanish in any point of $\Gamma_2$, so we can now complete the proof as in Lemma \ref{Lem:ExtremeRaysC4}.
\end{proof}

\begin{Rem}
In particular, the union of all extreme rays of $\Sigma_{2d}^\vee$ need not be closed, e.g.~for $d=9$, extremality fails in our original construction but a perturbation gives an extreme ray. 
\end{Rem}

\begin{Thm}\label{Thm:ZarClExtRays}
For any $d\geq 4$, the Zariski closure of the set of extreme rays of $\Sigma_{2d}^\vee$ is the variety of Hankel matrices of corank at least $4$. It is irreducible and has codimension $10$.
\end{Thm}

\begin{proof}
We have shown in the proof of Lemma \ref{Lem:Corank4dim} that the quasi-projective variety $\gor(T)$ of all Gorenstein ideals with Hilbert function $T(j)=\binom{j+2}{2}$ for $0\leq j<d$ and $T(d)=\binom{d+2}{2}-4$ is dense in $X_{-4}$. It is also smooth, cf.~Theorem \ref{Thm:GorTTangentSpace} or Iarrobino-Kanev \cite[Theorem 4.21]{IarKanMR1735271}. We have shown in Lemma \ref{Lem:ExtremeRaysC4} that there is an extreme ray $\R_+\ell_0$ of $\Sigma_{2d}^\vee$ with $I(\ell_0)\in\gor(T)$. We will now show that every linear functional in an open neighbourhood of $\ell_0$ in $\gor(T)$ spans an extreme ray of $\Sigma_{2d}^\vee$. Since $I(\ell)\in\gor(T)$ implies that the corank of the Hankel matrix $B_\ell$ is $4$, there is an open neighbourhood of $\ell_0$ such that $B_\ell$ is positive semi-definite for all $\ell$ in this neighbourhood, because the eigenvalues of a symmetric matrix depend continuously on its entries.
Therefore, a linear functional $\ell$ in this neighbourhood spans an extreme ray of $\Sigma_{2d}^\vee$ if and only if $I(\ell)_d$ generates a hyperplane in $\R[\ul{x}]_{2d}$, i.e.~$\langle I(\ell)_d\rangle_{2d}=I(\ell)_{2d}$.
By Gauss' algorithm (column echelon form), we can write a basis $(b_1,b_2,b_3,b_4)$ of the kernel of $B_\ell$ in terms of rational functions in the entries of $B_\ell$. We consider the linear map
\[
\R[\ul{x}]_d^4\to \R[\ul{x}]_{2d}, (f_1,f_2,f_3,f_4)\mapsto f_1b_1+f_2b_2+f_3b_3+f_4b_4.
\]
The rank of this map is at least $\binom{2d+2}{2}-1$ (i.e.~the image is a hyperplane) because $\ell\in\gor(T)$. The image is a hyperplane for $\ell=\ell_0$. So the same is true for every $\ell$ in a neighbourhood of $\ell_0$ in $\gor(T)$, which shows that these $\ell$ are extreme rays of $\Sigma_{2d}^\vee$.
\end{proof}

\begin{Rem}
In the proof of the above Theorem, we see that if $T$ is a Hilbert function occuring for a Gorenstein ideal corresponding to an extreme ray of $\Sigma_{2d}^\vee$, then there is an open subset of extreme rays in a connected component of $\gor(T)(\R)$ because $\gor(T)$ is smooth. As we remarked above, it might not be the entire connected component.
\end{Rem}

Our construction of an extreme ray of maximal rank also gives a base-point free special linear system with a totally real representative on a smooth curve of degree $d\geq 4$, which might be interesting in itself.
\begin{Prop}
Let $d\geq 4$. There is a smooth real curve $X\subset\P^2$ of degree $d$ and an effective divisor $D$ of degree $g = \binom{d-1}{2}$ supported on $X(\R)$ such that $|D|$ has dimension $1$ and is base-point free.
\end{Prop}

\begin{proof}
Start with a complete intersection $\V(L_1)\cap\V(L_2)$ of linear forms and a choice of $\binom{d-1}{2}$ points $\Gamma_2\subset\V(L_1)\cap\V(L_2)$ such that there is a unique curve of degree $d-3$ passing through these points. Moreover, assume that all coefficients in the linear relation among the point evaluations $\{\ev_v\in\R[\ul{x}]_{d-3}^\ast\colon v\in\Gamma_2\}$ are non-zero. This situation is established in the proof of Lemma \ref{Lem:ExRayC4Per}. By Bertini's Theorem \cite[Theorem 6.2.11]{BelCarMR2549804} or \cite[Th\'eor\`eme 6.6.2]{JouMR725671}, there is a smooth curve $\V(f)$ of degree $d$ passing through $\Gamma_2$ such that $f$ is a small perturbation of $L_1$; more precisely, we want $\Gamma = \V(f)\cap \V(L_2)$ to be a totally real transversal intersection. Then the complete linear system $|\Gamma_2|\subset \divi(\V(f))$ is cut out by forms of degree $d$, i.e.
\[
|\Gamma_2| = \{C.\V(f)-(\Gamma-\Gamma_2)\geq 0\colon C\subset\P^2\text{ of degree }d\},
\]
cf.~Eisenbud-Green-Harris \cite[Corollary 5 (to Brill-Noether's Restsatz)]{EisGreHar}. We have argued in Remark \ref{Rem:CBApplication} that this linear system is base-point free. We compute its dimension with the help of the Cayley-Bacharach Theorem, more precisely \cite[Corollary 6]{EisGreHar}:
\[
1 =|\Gamma_2|-( \ell( (d-3)H) - \ell( (d-3)H-\Gamma_2) )= g-( g-\ell( (d-3)H-\Gamma_2)),
\]
which implies
\[
\ell(\Gamma_2) = \deg(\Gamma_2) +1 - g +\ell( (d-3)H-\Gamma_2) = 2.
\]
\end{proof}

\begin{Rem}
Conversely, given such a linear system on a smooth curve $X\subset\P^2$, we can apply the construction in the proof of Lemma \ref{Lem:ExtremeRaysC4} to construct an extreme ray of $\Sigma_{2d}^\vee$ of maximal rank, at least if there is a totally real transveral intersection $C\cap X$ with $C.X-D\geq0$. The fact, that the linear system has dimension $1$ gives the unique linear relation among the point evaluations at $C.X-D$ on forms of degree $d$. Extremality then follows from the fact that $|D|$ is base-point free by the count of dimensions as in the proof of Lemma \ref{Lem:ExtremeRaysC4}.
\end{Rem}

\section{The case $d=5$ or Ternary Dextics.}\label{sec:Dextics}
For $d=3$, a complete characterisation of extreme rays of $\Sigma_6^\vee$ was given by Blekherman in \cite{BleNP}. It led to a complete description of the algebraic boundary of the sums of squares cone $\Sigma_6$ by Blekherman, Hauenstein, Ottem, Ranestad and Sturmfels, cf.~\cite{BleHauOttRanStuMR2999301}. For $d=4$, there are only two possible ranks for extreme rays of $\Sigma_8^\vee$, namely $10$ and $11$; in particular, we know how to construct one of each rank. It is possible to prove, similarly to the cases below, that both these ranks give rise to irreducible components of the algebraic boundary of $\Sigma_8$ by projective duality.

So the first interesting case from this point of view is $d=5$: In fact, we can construct an extreme ray of $\Sigma_{10}^\vee$ of every rank in the interval $\{13,\ldots,17\}$ between the lower and upper bound using the Cayley-Bacharach Theorem. Moreover, using the results of \cite{SinnAlgBound}, we can prove by projective duality that there is an irreducible component of the algebraic boundary of $\Sigma_{10}$ for every one of these ranks; in particular, $\partial_a\Sigma_{10}$ has at least $6$ irreducible components. In the following propositions in this section, we will prove the following theorem.
\begin{Thm}\label{Thm:d5}
For every $r\in\{13,\ldots,17\}$, there is an extreme ray $\R_+\ell_r$ of $\Sigma_{10}^\vee$ such that the rank of the Hankel matrix $B_{\ell_r}$ is $r$. The Hilbert function $T_r$ of $I(\ell_r)$ is
\begin{compactitem}
\item $T_{13} = (1,3,6,9,12,13,12,9,6,3,1)$ \\
\item $T_{14} = (1,3,6,10,13,14,13,10,6,3,1)$ \\
\item $T_{15} = (1,3,6,10,14,15,14,10,6,3,1)$ \\
\item $T_{16} = (1,3,6,10,14,16,14,10,6,3,1)$ \\
\item $T_{17} = (1,3,6,10,15,17,15,10,6,3,1)$ 
\end{compactitem}
The dual varieties to $\gor(T_r)$ are irreducible components of the algebraic boundary of the sums of squares cone $\Sigma_{10}$ for all $r\in\{13,\ldots,17\}$.
\end{Thm}

The construction given in the preceding section for extreme rays of maximal rank $\binom{d+2}{2}-4$ leads to an extreme ray $\R_+\ell$ of $\Sigma_{10}^\vee$ such that the Hilbert function of the corresponding Gorenstein ideal $I(\ell)$ is
\[
T_{17} = (1,3,6,10,15,17,15,10,6,3,1).
\]
By Theorem \ref{Thm:ZarClExtRays}, the Zariski closure of the set of extreme rays of $\Sigma_{10}^\vee$ is $\cl(\gor(T_{17}))$, a unirational variety of codimension $10$ in $\P^{65}$. So \cite[Theorem 3.8]{SinnAlgBound}, implies that its dual variety is an irreducible component of the algebraic boundary of $\Sigma_{10}$.

We now work our way up beginning with the lowest rank $13$, following the construction in Blekherman \cite{BlePositiveGorensteinIdeals}:
\begin{Prop}
There is an extreme ray $\R_+\ell$ of $\Sigma_{10}^\vee$ of rank $13$. The Hilbert function of the Gorenstein ideal $I(\ell)$ is
\[
T_{13}=(1,3,6,9,12,13,12,9,6,3,1)
\]
and the variety dual to $\cl(\gor(T_{13}))$ is an irreducible component of the algebraic boundary of $\Sigma_{10}$.
\end{Prop}

\begin{proof}
Let $L_1 = x(x-z)(x-2z)(x-3z)(x-4z)$ and $L_2 = y(y-z)(y-2z)$ and $\Gamma = \V(L_1)\cap\V(L_2)$. By construction, there is a unique linear relation among $\{\ev_v\in\R[\ul{x}]_5^\ast\colon v\in\Gamma\}$, say $\sum_{v\in\Gamma}u_v\ev_v = 0$, and all coefficients in this relation are non-zero. The linear functional
\[
\ell = \sum_{v\in\Gamma\setminus\{P\}} ev_v - \frac{u_P^2}{\sum_{v\in\Gamma\setminus\{P\}} u_v^2} \ev_P
\]
is positive semi-definite of rank $13$ for any $P\in\Gamma$ by the Cauchy-Schwarz inequality, cf.~proof of Lemma \ref{Lem:ExtremeRaysC4}. By a Hilbert function computation using \texttt{Macaulay2} \cite{Macaulay2}, we verify, that the degree $5$ part of the corresponding Gorenstein ideal $I(\ell)$ generates a hyperplane in degree $10$. To prove that the dual variety to $\gor(T_{13})$ is an irreducible component of $\partial_a \Sigma_{10}$, we use \cite[Theorem 3.8]{SinnAlgBound}. The condition given there is equivalent to
\[
(T_{\ell} \gor(T_{13}))^\perp = (I(\ell)_5)^2
\]
because the face of $\Sigma_{10}$ supported by $\ell$ is the set of sums of squares of polynomials in $I(\ell)_5$, which spans the vector space $(I(\ell)_5)^2$. By the description of the tangent space to $\gor(T_{13})$ at $\ell$ (cf.~Theorem \ref{Thm:GorTTangentSpace}), this is equivalent to
\[
(I(\ell)^2)_{10} = (I(\ell)_5)^2,
\]
which we also check using \texttt{Macaulay2} \cite{Macaulay2}.
\end{proof}

\begin{Prop}
There is an extreme ray $\R_+\ell$ of $\Sigma_{10}^\vee$ of rank $14$. The Hilbert function of the Gorenstein ideal $I(\ell)$ is
\[
T_{14}=(1,3,6,10,13,14,13,10,6,3,1)
\]
and the variety dual to $\cl(\gor(T_{14}))$ is an irreducible component of the algebraic boundary of $\Sigma_{10}$.
\end{Prop}

\begin{proof}
In this case, take $L_1 = x(x-z)(x-2z)(x-3z)$ and $L_2 = y(y-z)(y-2z)(y-3z)$ and set $\Gamma = \V(L_1)\cap \V(L_2)$. There is a unique linear relation among $\{\ev_v\in\R[\ul{x}]_5^\ast \colon v\in\Gamma\}$, say $\sum_{v\in\Gamma} u_v\ev_v = 0$, and all its coefficients are non-zero. As above, the linear functional
\[
\ell = \sum_{v\in\Gamma\setminus\{P\}} ev_v - \frac{u_P^2}{\sum_{v\in\Gamma\setminus\{P\}} u_v^2} \ev_P
\]
is positive semi-definite of rank $14$ for any $P\in\Gamma$. Again, using \texttt{Macaulay2} \cite{Macaulay2}, we verify, that the degree $5$ part of the corresponding Gorenstein ideal $I(\ell)$ generates a hyperplane in degree $10$ and that
\[
(I(\ell)^2)_{10} = (I(\ell)_5)^2.
\]
\end{proof}

\begin{Prop}
There is an extreme ray $\R_+\ell$ of $\Sigma_{10}^\vee$ of rank $15$. The Hilbert function of the Gorenstein ideal $I(\ell)$ is
\[
T_{15}=(1,3,6,10,14,15,14,10,6,3,1)
\]
and the variety dual to $\cl(\gor(T_{15}))$ is an irreducible component of the algebraic boundary of $\Sigma_{10}$.
\end{Prop}

\begin{proof}
In this case, we start with a complete intersection of a quartic and a quintic, $L_1 = x(x-z)(x-2z)(x-3z)(x-4z)$, $L_2 = y (y-z)(y-2z)(y-3z)$ and $\Gamma = \V(L_1)\cap \V(L_2)$. Choose $\Gamma_2 = \{(0:2:1),(1:1:1),(2:0:1)\}$ and set $\Gamma_1 = \Gamma\setminus\Gamma_2$. By Cayley-Bacharach, there is a unique linear relation among the $17$ points of $\Gamma_1$. Using \texttt{Macaulay2} \cite{Macaulay2}, we complete the proof as above.
\end{proof}

\begin{Prop}
There is an extreme ray $\R_+\ell$ of $\Sigma_{10}^\vee$ of rank $16$. The Hilbert function of the Gorenstein ideal $I(\ell)$ is
\[
T_{16}=(1,3,6,10,14,16,14,10,6,3,1)
\]
and the variety dual to $\cl(\gor(T_{16}))$ is an irreducible component of the algebraic boundary of $\Sigma_{10}$.
\end{Prop}

\begin{proof}
Again, choose $L_1 = x(x-z)(x-2z)(x-3z)(x-4z)$, $L_2 = y (y-z)(y-2z)(y-3z)$ and $\Gamma = \V(L_1)\cap \V(L_2)$. This time, $\Gamma_2 = \{(0:1:1),(1:0:1)\}$ and $\Gamma_1 = \Gamma\setminus\Gamma_2$ do the job: Cayley-Bacharach gives a unique linear relation among the $18$ points of $\Gamma_1$. Using \texttt{Macaulay2} \cite{Macaulay2}, we complete the proof as above.
\end{proof}

For general $d>5$, our constructive method using the Cayley-Bacharach Theorem cannot construct an extreme ray of every rank in the interval $\{3d-2,\ldots,\binom{d+2}{2}\}$ given by the lower and upper bound. The first failure occurs for $d=6$ and rank $17\in\{16,\ldots,24\}$. In fact, $\Sigma_{12}^\vee$ does not have an extreme ray of rank $17$, as we will see below, cf.~Lemma \ref{Lem:Rank17}.
\begin{Rem}\label{Rem:d6r17}
Our construction starts with a totally real intersection of two curves $X_1$ and $X_2$ with $\deg(X_1)+\deg(X_2)\geq d+3$; we then need $19$ intersection points such that the corresponding point evaluations on forms of degree $6$ satisfy a unique linear relation in which all coefficients are non-zero. This configuration would lead to a positive linear functional such that the Hankel matrix has the desired rank $17$ (of course we would still need to prove extremality). We will see that this is not possible:

The following tuples are permissible choices for the degrees of the curves $(3,6)$, $(4,5)$, $(4,6)$, $(5,5)$, $(5,6)$ and $(6,6)$. 
For $(\deg(X_1),\deg(X_2)) = (3,6)$, the transversal intersection has only $18$ points. In the case $(4,5)$, there is a unique linear relation among point evaluations at the $20$ intersection points such that all coefficients are non-zero; in particular, whatever point we remove, the remaining $19$ point evaluations are linearly independent on forms of degree $6$. In the cases $(4,6)$, $(5,5)$ and $(5,6)$, we cannot have the desired number of points on a curve of dual degree $s-d$: For example, in order to apply the duality of the Cayley-Bacharach Theorem to the $24$ intersection points in the case $(4,6)$, we would need to have $5$ of the intersection points on a line, which intersects the quartic in only $4$ points. The last case $(6,6)$ is more subtle: We would like to find exactly $17$ intersection points on a cubic, which is impossible, because there is a unique linear relation among the corresponding point evaluations on forms of degree $6$ on the complete intersection of a cubic and a sextic, cf.~Eisenbud-Green-Harris \cite[CB4]{EisGreHar}.
\end{Rem}

This is not a defect of our construction in this case. In fact, there are no extreme rays of $\Sigma_{12}^\vee$ of rank $17$.
\begin{Lem}\label{Lem:Rank17}
There is no Gorenstein ideal $I\subset \C[x,y,z]$ with socle in degree $12$ such that $\hilb(I,6) = 17$ and $I_6$ is maximal with respect to inclusion among $J_6$, where $J$ runs over all Gorenstein ideals with socle in degree $12$.
\end{Lem}

In the proof of this lemma, we will use the following theorem multiple times.
\begin{Rem}
The complete intersection of three ternary forms of degree $d_1$, $d_2$, and $d_3$ respectively is a Gorenstein ideal in $\C[x,y,z]$ with socle in degree $d_1+d_2+d_3-3$, see \cite[Theorem CB8]{EisGreHar}. The socle degree follows using an elementary count of dimensions and the fact that the generators must be relatively prime.
\end{Rem}

\begin{proof}
We exclude possibilities arguing by the lowest degree $k$ of a generator of $I$. First note that maximality of $I_6$ implies that $\langle I_6 \rangle_{12} = I_{12}$ and $\V(I_6)=\emptyset$. In particular, we can always choose a complete intersection of three forms in $I_6$, one of which can be chosen to be a suitable multiple of a generator of minimal degree of $I$. Let $k$ be the minimal degree of a generator of $I$. The ideal $I$ cannot contain a quadric generator, because the linear function defining $I$ would then be supported on $12$ points by the apolarity lemma, which implies $\hilb(I,6) \leq 12$. In case $k=3$, the Gorenstein ideal is actually generated by a cubic and two sextics that are a complete intersection, see Stanley \cite{Stanley}, so $\hilb(I,6) = 16 = 28 - (10 +2)$. The case $k=6$ is also easily excluded because Hilbert functions of Gorenstein ideals in $\C[x,y,z]$ are unimodular by \cite[Theorem 4.2]{Stanley}, which implies in this case that $\hilb(I,5)\leq \hilb(I,6) = 17$, or equivalently $\dim(I_5) \geq 4$. This leaves the two cases $k=5$ and $k=4$: 

Suppose $k=5$, then the Hilbert functions of Gorenstein ideals with socle in degree $12$ and order $5$ are in 1-1 correspondence with self-complimentary partitions of $10\times 4$ blocks, cf.~\cite[Proposition 3.9]{Die}. By unimodularity of Hilbert functions, we have $\dim(I_5)\geq 4$. This determines the first three rows of the blocks, so we can choose two more generators of degree $\leq 6$. Since a block of degree $5$ forces a relation in degree $6$ by the self-complimentarity of the partition, the $4$ generators of degree $5$ generate a $4\cdot 3 - 3 = 9$-dimensional subspace of $\C[x,y,z]_6$. The only two possible degrees for the other two generators to achieve $\dim(I_6) = 28-17 = 11$ are therefore one more generator of degree $5$ and one generator of degree $7$ or two generators in degree $6$. In the first case, the degrees of generators are $q = (5,5,5,5,5,7,7,9,9,9,9)$ and the corresponding relation degrees are $p = (10,10,10,10,10,8,8,6,6,6,6)$. Since there is no generator of degree $6$ and $\V(I_6) = \emptyset$, we find a complete intersection of three forms of degree $5$ contained in $I$. These generate a Gorenstein ideal with socle in degree $5+5+5-3 = 12$. This is impossible, becaues $I$ contains further generators. In the second case, the degrees of generators are $q = (5,5,5,5,6,6,8,8,9,9,9)$ and the corresponding relation degrees are $p = (10,10,10,10,9,9,7,7,6,6,6)$. The quasiprojective variety $\gor(T)$, where $T$ is given by these generator and relation degrees, contains a dense subset of Gorenstein ideals generated by polynomials of degree $q_{\rm min} = (5,5,5,5,8,8,9)$ by \cite[Section 3.3]{Die}. So the same argument as in the first case excludes this possibility, too.

So now we are left with the case $k=4$: In this case, Hilbert functions of Gorenstein ideals with socle in degree $12$ and order $4$ correspond to self-complimentary partitions of $8\times 6$ blocks. If $\dim(I_4) =2$, then these would generate a $2\cdot 6 = 12$-dimensional subspace in $\C[x,y,z]_6$ because they correspond to relations in degree $7$. So $\dim(I_4) = 1$. We can choose $4$ more degrees of generators $\leq 6$. A generator of degree $5$ comes with a relation in degree $5$ and therefore, the only two possible choices of degrees for the generators with $\hilb(I,6) = 17$ are one generator of degree $5$ and three generators of degree $6$ or two generators of degree $5$ and one generator of degree $6$. Let's first consider $q = (4,5,5,6,7,7,8,9,9)$, which corresponds to relation degrees $p = (11,10,10,9,8,8,7,6,6)$. Again, $\gor(T)$, where $T$ is given by these generator and relation degrees, contains a dense subset of Gorenstein ideals generated by polynomials of degree $q_{\rm min} = (4,5,5,7,9)$. There is no generator of degree $6$ anymore, so $\V(I_6) = \emptyset$ implies that we find a complete intersection of a quartic and two quintics, which generate a Gorenstein ideal with socle in degree $4+5+5-3=11$, which is impossible. The last remaining case is $q = (4,5,6,6,6,8,8,8,9)$ with corresponding relation degrees $p = (11,10,9,9,9,7,7,7,6)$. Here, $\gor(T)$ contains a dense subset of Gorenstein ideals with generators of degree $q_{\rm min} = (4,5,6,6,8,8,8)$ and correspondingly $p_{\rm min} = (11,10,9,9,7,7,7)$. The assumption $\V(I_6) = \emptyset$ implies only that we can find a complete intersection of a quartic and two sextics. They generate a Gorenstein ideal with socle in degree $4+6+6-1 = 13$. This Gorenstein ideal has Hilbert function $3$ in degree $12$ because Hilbert functions of Gorenstein ideals are symmetric, cf.~Remark \ref{Rem:SymHilbFunc}. This is impossible because this complete intersection together with the other $4$ generators would then fill $\C[x,y,z]_{12}$. This concludes the case study.
\end{proof}


\textbf{Acknowledgements.} This paper grew out of the PhD thesis of the second author, who wants to thank his adviser Claus Scheiderer for his support, ideas and input. 
The first author was partially supported by Alfred P. Sloan research fellowship and NSF DMS CAREER award. The second author was supported by the Studienstiftung des deutschen Volkes and the National Institute of Mathematical Sciences, Daejeon, Korea, during the Summer 2014 Thematic Program on {\em Applied Algebraic Geometry}.

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1406.1873.extracted_bib
{BelCarMR2549804 Mauro~C. Beltrametti, Ettore Carletti, Dionisio Gallarati, and Giacomo Monti~Bragadin. \newblock {\em Lectures on curves, surfaces and projective varieties, a classical view of algebraic geometry. \newblock EMS Textbooks in Mathematics. European Mathematical Society (EMS), Z\"urich, 2009. Translated from the 2003 Italian original by Francis Sullivan.
1406.1873.extracted_bib
{BlePositiveGorensteinIdeals Grigoriy Blekherman. \newblock Positive Gorenstein ideals. \newblock to appear in Proceedings of the AMS.
1406.1873.extracted_bib
{BleNP Grigoriy Blekherman. \newblock Nonnegative polynomials and sums of squares. \newblock {\em J. Amer. Math. Soc., 25(3):617--635, 2012.
1406.1873.extracted_bib
{BleHauOttRanStuMR2999301 Grigoriy Blekherman, Jonathan Hauenstein, John~Christian Ottem, Kristian Ranestad, and Bernd Sturmfels. \newblock Algebraic boundaries of {Hilbert's {SOS cones. \newblock {\em Compos. Math., 148(6):1717--1735, 2012.
1406.1873.extracted_bib
{BleParThoMR3075433 Grigoriy Blekherman, Pablo~A. Parrilo, and Rekha~R. Thomas, editors. \newblock {\em Semidefinite optimization and convex algebraic geometry, volume~13 of {\em MOS-SIAM Series on Optimization. \newblock Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2013.
1406.1873.extracted_bib
{BochnakMR1659509 Jacek Bochnak, Michel Coste, and Marie-Fran{\c{coise Roy. \newblock {\em Real algebraic geometry, volume~36 of {\em Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. \newblock Springer-Verlag, Berlin, 1998. \newblock Translated from the 1987 French original, Revised by the authors.
1406.1873.extracted_bib
{BE David~A. Buchsbaum and David Eisenbud. \newblock Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension {$3$. \newblock {\em Amer. J. Math., 99(3):447--485, 1977.
1406.1873.extracted_bib
{Die Susan~J. Diesel. \newblock Irreducibility and dimension theorems for families of height {$3$ {Gorenstein algebras. \newblock {\em Pacific J. Math., 172(2):365--397, 1996.
1406.1873.extracted_bib
{EisenbudMR1322960 David Eisenbud. \newblock {\em Commutative algebra, with a view toward algebraic geometry, volume 150 of {\em Graduate Texts in Mathematics. \newblock Springer-Verlag, New York, 1995.
1406.1873.extracted_bib
{EisGreHar David Eisenbud, Mark Green, and Joe Harris. \newblock Cayley-Bacharach theorems and conjectures. \newblock {\em Bull. Amer. Math. Soc, 33(3):295--324, 1996.
1406.1873.extracted_bib
{Macaulay2 Daniel Grayson and Michael Stillman. \newblock Macaulay2, a software system for research in algebraic geometry. \newblock available at http://www.math.uiuc.edu/Macaulay2/.
1406.1873.extracted_bib
{HilMR1510517 David Hilbert. \newblock Ueber die {Darstellung definiter {Formen als {Summe von {Formenquadraten. \newblock {\em Math. Ann., 32(3):342--350, 1888.
1406.1873.extracted_bib
{IarKanMR1735271 Anthony Iarrobino and Vassil Kanev. \newblock {\em Power sums, {Gorenstein algebras, and determinantal loci, volume 1721 of {\em Lecture Notes in Mathematics. \newblock Springer-Verlag, Berlin, 1999. \newblock Appendix C by Iarrobino and Steven L. Kleiman.
1406.1873.extracted_bib
{JouMR725671 Jean-Pierre Jouanolou. \newblock {\em Th\'eor\`emes de {Bertini et applications, volume~42 of {\em Progress in Mathematics. \newblock Birkh\"auser Boston Inc., Boston, MA, 1983.
1406.1873.extracted_bib
{Reznick Bruce Reznick. \newblock On Hilbert's construction of positive polynomials. \newblock {\tt arXiv:0707.2156
1406.1873.extracted_bib
{SinnAlgBound Rainer Sinn. \newblock Algebraic boundaries of convex semi-algebraic sets. \newblock {\tt arXiv:1405.7822.
1406.1873.extracted_bib
{Stanley Richard P. Stanley. \newblock Hilbert functions of graded algebras, \newblock {\em Advances in Math., 28(1):57--83, 1978.