Spaces of Sums of Powers and Real Rank Boundaries

Abstract

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 $\mathrm{SSP}$ to various loci in the Hilbert scheme of four points in ${\mathbb{P}}^3$. 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 $J({\sigma }_3(v_3({\mathbb{P}}^3)),\tau (v_3({\mathbb{P}}^3)))$.

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• Finite schemes and secant varieties over arbitrary characteristic