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Secant varieties of Segre-Veronese varieties \(\mathbb P^m \times \mathbb P^n\) embedded by \(\mathcal O(1,2)\). (English) Zbl 1198.14051

Consider the embedding \(X_{m,n}\subset {\mathbb P}^N\) of \({\mathbb P}^n\times {\mathbb P}^m\) into \({\mathbb P}^N\), \(N=(m+1)\times {n+2\choose 2}-1\), defined by \({\mathcal O}(1,2)\); the present article studies the dimension of the secant varieties \(\sigma_s(X_{m,n})\), which are defined as the closure of the union of all linear spaces spanned by \(s\) points of \( X_{m,n}\).
The variety \(X_{m,n}\) is a particular example of a Segre-Veronese variety (the embedding of a product of projective spaces via the linear system of pluri-homogeneous polynomial of a given multi-degree). Such secant varieties have been studied by many authors, and their interest lies also in the fact that they parameterize particular partially symmetric tensors.
It is not known in general which Segre-Veronese varieties have some secant variety of unexpected dimension (defective secant varieties), not even for the ones considered here; only for Veronese varieties a complete description of the defective cases has been given (by Alexander-Hirschowitz theorem).
In this paper, via the use of Castelnuovo sequence and a clever induction, it is shown that \(\sigma_s(X_{m,n})\) has the expected dimension for all \(s\leq \underline s(m,n)\) and \(s\geq \overline s(m,n)\), for two appositly defined functions \(\underline s(m,n)\) and \( \overline s(m,n)\).
In the last section a conjecture is given which describes which are the only cases when \(\sigma_s(X_{m,n})\) should be defective (all those cases are known to be defective). Such conjecture has been tested by computer for all \(m,n \leq 10\).

MSC:

14M99 Special varieties
14Q99 Computational aspects in algebraic geometry
15A69 Multilinear algebra, tensor calculus
15A72 Vector and tensor algebra, theory of invariants

Software:

Macaulay2; SeDiMO
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