Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023
Edoardo Ballico ^{[a]}
Typical labels of real forms
Pages: 8795
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)
Full Text (PDF)
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 labelweight \(t\) is typical for the \(k\)secant variety
\(\sigma _k(X(\mathbb{C}))\) of \(X(\mathbb{C})\) if there is a nonempty 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 labelweight of \(\sigma _k(X(\mathbb{C}))(\mathbb{R})\) if \(\sigma _{k1}(X(\mathbb{C}))\ne \mathbb{P}^r(\mathbb{C})\).
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 labelweights 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 AlexanderHirschowitz
list of exceptional secant varieties of Veronese embeddings.
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