Edoardo Ballico ^[a]

Typical labels of real forms

Pages: 87-95
Received: 28 May 2022
Accepted in revised form: 17 January 2023
Mathematics Subject Classification: 14N05; 15A69
Keywords: Real homogenous polynomial; additive decomposition of polynomials; typical rank
Author address:
[a]: University of Trento, Dept. of Mathematics, 38123 Povo (TN), Italy

The author was partially supported by MIUR and GNSAGA of INdAM (Italy)

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Abstract: Let \(X(\mathbb{C})\subset \mathbb{P}^r(\mathbb{C})\) be an integral projective variety defined over \(\mathbb{R}\). Let \(\sigma\) denote the complex conjugation. A point \(q\in \mathbb{P}^r(\mathbb{R})\) is said to have \((a,b)\in \mathbb{N}^2\) as a label if there is \(S\subset X(\mathbb{C})\) such that \(\sigma(S)=S\), \(S\) spans \(q\), \(\#S =2a+b\) and \(\#(S\cap X(\mathbb{R})) =b\). We say that \((a,b)\) has weight \(2a+b\). A label-weight \(t\) is typical for the \(k\)-secant variety \(\sigma _k(X(\mathbb{C}))\) of \(X(\mathbb{C})\) if there is a non-empty euclidean open subset \(V\) of \(\sigma _k(X(\mathbb{C}))(\mathbb{R})\) such that all \(q\in V\) have a label of weight \(t\) and no label of weight \( < t\).The integer \(k\) is always the minimal label-weight of \(\sigma _k(X(\mathbb{C}))(\mathbb{R})\) if \(\sigma _{k-1}(X(\mathbb{C}))\ne \mathbb{P}^r(\mathbb{C})\). In this paper \(X(\mathbb{C}) =X_{n,d}(\mathbb{C})\) is the order \(d\) Veronese embedding of \(\mathbb{P}^{n}(\mathbb{C})\). We prove that \(k\) and \(k+1\) are the typical label-weights of \(\sigma _k(X(\mathbb{C}))(\mathbb{R})\) if \((n,d,k)\in \{(2,6,9),(3,4,8),(5,3,9),(2,4,5),(4,3,7)\}\). These examples are important, because the first \(3\) are the ones in which generic uniqueness for proper secant varieties fails for the \(k\)-secant variety (a theorem by Chiantini, Ottaviani and Vannieuwenhoven), the fourth is in the Mukai list (fano \(3\)-fold \(V_{22}\)) and the last one appears in the Alexander-Hirschowitz list of exceptional secant varieties of Veronese embeddings.