Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited

2007/02/06 by Giorgio Ottaviani, Ottaviani, Giorgio · 1 citation
Mathematics · Physics and Astronomy · #14F05 #14H50 #14J60 #14N15 #15A72 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.math/0702151

openalex publication_date 2007/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X=\bf P²×\bf P^n-1 embedded with Ø(1,2). We prove that its (n+1)-secant variety σ_n+1(X) is a hypersurface, while it is expected that it fills the ambient space. The equation of σ_n+1(X) is the symmetric analog of the Strassen equation. When n=4 the determinantal map takes σ₅(X) to the hypersurface of Lüroth quartics, which is the image of the Barth map studied by LePotier and Tikhomirov. This hint allows to obtain some results on the jumping lines and the Brill-Noether loci of symplectic bundles on \bf P² by using the higher secant varieties of X.

Citations

• On the ideals and singularities of secant varieties of Segre varieties

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