Spaces of Sums of Powers and Real Rank Boundaries

We investigate properties of Waring decompositions of real homogeneous forms. We study the moduli of real decompositions, so-called Space of Sums of Powers, naturally included in the Variety of Sums of Powers. Explicit results are obtained for quaternary quadrics, relating the algebraic boundary of ${\rm SSP}$tovariouslociintheHilbertschemeoffourpointsin$$\mathbb{P}$^3$. Further, we study the locus of general real forms whose real rank coincides with the complex rank. In case of quaternary quadrics the boundary of this locus is a degree forty hypersurface $J({\sigma }_3(v_3({\mathbb{P}}^3)),\tau (v_3({\mathbb{P}}^3)))$.