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Real rank geometry of ternary forms. (English) Zbl 1390.14179

The main of this extremely interesting paper is to understand real ternary forms whose real rank is equal to the generic complex rank.
Let us denote by \(\mathbb{R}[x,y,z]_{d}\) the \({d+2 \choose 2}\)-dimensional vector space of ternary forms \(f\) of degree \(d\) which are homogeneous polynomials of degree \(d\) in \(x,y,z\), or equivalently symmetric tensors of format \(3 \times 3 \times \cdots \times 3\) with \(d\) factors. One would like to understand decompositions \[ (\triangle): \quad \quad f(x,y,z) = \sum_{i=1}^{r}\lambda_{i}(a_{i}x+b_{i}y+c_{i}z)^{d}, \] where \(\lambda_{i},a_{i},b_{i},c_{i} \in \mathbb{R}\) with \(i \in \{1,\dots,r\}\). The smallest \(r\) for which such a representation exists is the real rank of \(f\), denoted by \(\text{rk}_{\mathbb{R}}(f)\). The complex rank \(\text{rk}_{\mathbb{C}}(f)\) is the smallest \(r\) such that \(f\) has the form \((\triangle)\) with \(a_{i},b_{i},c_{i} \in \mathbb{C}\). The inequality \(\text{rk}_{\mathbb{C}}(f) \leq \text{rk}_{\mathbb{R}}(f)\) always holds, and it is often strict. From now on we assume that \(f\) is a general ternary form in \(\mathbb{R}[x,y,z]_{d}\) The complex rank of such a form is referred to as the generic rank, and it depends only on \(d\), so it will be denoted by \(R(d)\). Let us define \[ \mathcal{R}_{d} = \{f \in \mathbb{R}[x,y,z]_{d} : \text{rk}_{\mathbb{R}}(f) = R(d)\}. \] This is a full-dimensional semialgebraic subset of \(\mathbb{R}[x,y,z]_{d}\). Its topological boundary \(\partial \mathcal{R}_{d}\) is the set-theoretic difference of the closure of \(\mathcal{R}_{d}\) minus the interior of the closure of \(\mathcal{R}_{d}\). Thus, if \(f \in \partial \mathcal{R}_{d}\), then every open neighborhood of \(f\) contains a general form of real rank equal to \(R(d)\) and also a general form of real rank bigger than \(R(d)\). The semialgebraic set \(\partial \mathcal{R}_{d}\) is either empty or pure of codimension \(1\). The real rank boundary \(\partial_{\mathrm{alg}}(\mathcal{R}_{d})\) is defined as the Zariski closure of the topological boundary \(\partial \mathcal{R}_{d}\) in the complex projective space \(\mathbb{P}(\mathbb{C}[x,y,z]_{d})\). The authors conjecture that the variety \(\partial_{\mathrm{alg}}(\mathcal{R}_{d})\) is non-empty and hence has codimension \(1\) for all \(d\geq 4\). The authors verify this conjecture for \(d=6,7,8\) providing the affirmative answer.
For any ternary form \(f\) and the generic rank \(R(d)=r\) it is natural to ask for the space of all decompositions \((\triangle)\). In algebraic geometry, we call this space the variety of sums of powers \(\text{VSP}(f)\). By definition, \(\text{VSP}(f)\) is the closure of the subscheme of the Hilbert scheme \(\text{Hilb}_{r}(\mathbb{P}^{2})\) parametrizing the unordered configurations \[ (*): \quad \quad\{(a_{1}:b_{1}:c_{1}); \cdots ;(a_{r}:b_{r}:c_{r})\} \subset \mathbb{P}^{2} \] that can occur in \((\triangle)\). If \(f\) is general, then the dimension of \(\text{VSP}(f)\) depends only on \(d\), and in two-thirds known cases the variety \(\text{VSP}(f)\) is finite. Another interesting problem that the authors deal with is devoted to the semialgebraic set \(\text{SSP}_{\mathbb{R}}(f)\) of those configurations \((*)\) in \(\text{VSP}(f)\) whose \(r\) points all have real coordinates. This is the space of real sums of powers, and \(\text{SSP}_{\mathbb{R}}(f)\) is non-empty iff the ternary form \(f\) lies in the semialgebraic set \(\mathcal{R}_{d}\).
Let us present some main results from the paper which are devoted to ternary forms of degree \(d \in \{2,3,\dots, 7\}\).
We start with quadrics. If \(f\) is a real quadratic form \(f\) in \(n\) variables of signature \((p,q)\), then after a linear change of coordinates \[ f = x_{1}^{2} + \cdots + x_{p}^{2} -x_{p+1}^{2} - \cdots - x_{p+q}^{2} \] with \(n=p+q\). The stabilizer of \(f\) in \(\mathrm{GL}(n,\mathbb{R})\) is denoted by \(\mathrm{SO}(p,q)\), and we denote by \(\mathrm{SO}^{+}(p,q)\) the connected component of \(\mathrm{SO}(p,q)\) containing the identity. Denote by \(G\) the stabilizer in \(\mathrm{SO}^{+}(p,q)\) of the set \(\{\{x_{1}^{2}, \cdots, x_{p}^{2}\},\{x_{p+1}^{2}, \cdots ,x_{p+q}^{2}\}\}\). In particular, if \(f\) is positive definite, then we get \(\mathrm{SO}^{+}(n,0) = \mathrm{SO}(n)\) and \(G\) is the subgroup of rotational symmetries of the \(n\)-cube.
Theorem 1. Let \(f\) be a rank \(n\) quadric of signature \((p,q)\). The space \(\text{SSP}_{\mathbb{R}}(f)\) can be identified with the quotient \(\mathrm{SO}^{+}(p,q)/G\). If the quadric \(f\) is definite, then \(\text{SSP}_{\mathbb{R}}(f) = \text{VSP}_{\mathbb{R}}(f) = \mathrm{SO}(n)/G\). In all other cases, \(\overline{\text{SSP}_{\mathbb{R}}(F)}\) is strictly contained in the real variety \(\text{VSP}_{\mathbb{R}}(f)\).
If \(f\) is a real general cubic, then one can prove the following.
Theorem 2. The semialgebraic set \(\text{SSP}_{\mathbb{R}}(f)\) is either a disk in the real projective plane or a disjoint union of a disk and a Möbius strip. The algebraic boundary of \(\text{SSP}_{\mathbb{R}}(f)\) is an irreducible sextic curve that has nine cusps.
For quartics the authors present the following description.
Theorem 3. The algebraic boundary \(\partial_{\mathrm{alg}}(\mathcal{R}_{4})\) is a reducible hypersurface in the \(\mathbb{P}^{14}\) of quartics. One of irreducible components has degree \(51\), and another irreducible component divides the region of hyperbolic quartics.
In the context of quartics, the authors formulate an interesting conjecture.
Conjecture. The real rank boundary \(\partial_{\mathrm{alg}}(\mathcal{R}_{4})\) for ternary quartics is a reducible hypersurface of degree \(84 = 6 + 27 + 51\), and it has exactly three irreducible components which can be described explicitely.
Theorem 4. Let \(f\) be a general ternary quartic of real rank \(6\) (the maximal value via Alexander-Hirschowitz Theorem). Using the affine coordinates \(v_{i,j}\) on \(\text{Gr}(4,7)\), the threefold \(\text{VSP}_{\mathbb{R}}(f)\) is defined by nine quadratic equations in \(\mathbb{R}^{12}\). If \(f\) has signature \((6,0)\), then \(\text{SSP}_{\mathbb{R}}(f)\) equals to \(\text{VSP}_{\mathbb{R}}(f)\). If \(\overline{ \text{SSP}_{\mathbb{R}}(f)}\) is a proper subset of \(\text{VSP}_{\mathbb{R}}(f)\), then its algebraic boundary has degree \(84\).
Now we focus on general ternary quintics and septics.
Theorem 5. The algebraic boundary \(\partial_{\mathrm{alg}}(\mathcal{R}_{5})\) of the set \(\mathcal{R}_{5} = \{f : \text{rk}_{\mathbb{R}}(f) = 7\}\) is an irreducible hypersurface of degree \(168\) in the \(\mathbb{P}^{20}\) of quintics. It has the parametric representation \[ g = l_{1}^{5} + \cdots + l_{5}^{5} + l_{6}^{4}l_{7} \] with \(l_{i}\)’s in \(\mathbb{R}[x,y,z]_{1}\).
Theorem 6. The real rank boundary \(\partial_{\mathrm{alg}}(\mathcal{R}_{7})\) is a non-empty hypersurface in \(\mathbb{P}^{35}\) with one of the components equal to the join of the tenth secant variety and the tangential variety.
Finally, in the case of sextics one can show the following.
Theorem 7. The algebraic boundary \(\partial_{\mathrm{alg}}(\mathcal{R}_{6})\) is a hypersurface in the \(\mathbb{P}^{27}\) of ternary sextics. One of its irreducible components is the dual to the Severi variety of rational sextics.

MSC:

14P10 Semialgebraic sets and related spaces
51N35 Questions of classical algebraic geometry
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