On the algebraic structure of the copositive cone

We decompose the copositive cone $\copos{n}$ into a disjoint union of a finite number of open subsets $S_{\cal E}$ of algebraic sets $Z_{\cal E}$. Each set $S_{\cal E}$ consists of interiors of faces of $\copos{n}$. On each irreducible component of $Z_{\cal E}$ these faces generically have the same dimension. Each algebraic set $Z_{\cal E}$ is characterized by a finite collection ${\cal E} = \{(I_{\alpha},J_{\alpha})\}_{\alpha = 1,\dots,|{\cal E}|}$ of pairs of index sets. Namely, $Z_{\cal E}$ is the set of symmetric matrices $A$ such that the submatrices $A_{I_{\alpha} \times J_{\alpha}}$ are rank-deficient for all $\alpha$. For every copositive matrix $A \in S_{\cal E}$, the index sets $I_{\alpha}$ are the minimal zero supports of $A$. If $u^{\alpha}$ is a corresponding minimal zero of $A$, then $J_{\alpha}$ is the set of indices $j$ such that $(Au^{\alpha})_j = 0$. We call the pair $(I_{\alpha},J_{\alpha})$ the extended support of the zero $u^{\alpha}$, and ${\cal E}$ the extended minimal zero support set of $A$. We provide some necessary conditions on ${\cal E}$ for $S_{\cal E}$ to be non-empty, and for a subset $S_{{\cal E}'}$ to intersect the boundary of another subset $S_{\cal E}$.