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

Abstract

Let $X={\mathbf{P}}^2\times {\mathbf{P}}^{n-1}$ embedded with $Ø(1,2)$. We prove that its $(n+1)$-secant variety ${\sigma }_{n+1}(X)$ is a hypersurface, while it is expected that it fills the ambient space. The equation of ${\sigma }_{n+1}(X)$ is the symmetric analog of the Strassen equation. When $n=4$ the determinantal map takes ${\sigma }_5(X)$ 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 ${\mathbf{P}}^2$ by using the higher secant varieties of $X$.

arXiv comment26 pages

References in corpus (2)

• On the ideals and singularities of secant varieties of Segre varieties
• Generalizations of Strassen's equations for secant varieties of Segre varieties

Cited by in corpus (9)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• Toward a salmon conjecture
• Four lectures on secant varieties
• A syzygetic approach to the smoothability of zero-dimensional schemes
• Branes in the moduli space of framed instantons
• On the hypersurface of Luroth quartics
• The fundamental invariants of 3 x 3 x 3 arrays
• Moduli of autodual instanton bundles
• On tensor rank and commuting matrices