On Waring's problem for systems of skew-symmetric forms (Q2435389)

The famous Waring problem for \(d\)-forms (i.e., homogeneous polynomials of degree \(d\)) asks which is the smallest \(s\) such that a general \(d\)-form in \(n+1\) variables can be expressed as sum of \(s\) \(d\)-th powers of linear forms. This question can be translated in terms of computing the dimension of the higher secant varieties to Veronese varieties of dimension \(n\) and degree \(d\). The solution of this problem is a consequence of the well-known Alexander-Hirschowitz theorem. Recently some natural variations of Waring problem have attracted the interest of many researchers. For example, one can ask which is the smallest \(s\) such that \(m+1\) general \(d\)-forms in \(n+1\) variables can be expressed as linear combinations of the same \(s\) \(d\)-th powers of linear forms. This is related to the computation of the dimension of higher secant varieties to Segre-Veronese varieties \(\mathbb{P}^m\times\mathbb{P}^n\) of bidegree \((1,d)\). The paper under review poses a similar question focusing on skew-symmetric forms instead of polynomials. More precisely, the authors ask which is the smallest \(s\) such that \(m+1\) general skew-symmetric \((k+1)\)-forms in \(n+1\) variables can be expressed as linear combinations of the same \(s\) decomposable skew-symmetric forms. The geometric counterpart of this problem is related to \textit{Grassmann secant varieties} of Grassmann varieties, and to the so-called \textit{Grassmann defectivity}, see [\textit{C. Fontanari} and \textit{C. Dionisi}, Matematiche 56, No. 2, 245--255 (2001; Zbl 1177.14093)]. A complete classification of the defective Grassmann secant varieties of Grassmann varieties would answer to the Waring problem in this setting. Hence the authors develop algebraic and geometrical tools to prove defectivity, and this allows them to find new relevant defective Grassmann secant varieties of Grassmann varieties. Note that an infinite family of defective cases, the \textit{unbalanced} ones, was previously known by \textit{E. Ballico} et al. [Linear Algebra Appl. 438, No. 1, 121--135 (2013; Zbl 1255.14044)] and \textit{J. Buczyński} and \textit{J. M. Landsberg} [Linear Algebra Appl. 438, No. 2, 668--689 (2013; Zbl 1268.15024)]. This paper presents a new infinite family and four sporadic cases of defective Grassmann secant varieties of Grassmann varieties. In each of these cases the defect is also computed.