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

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

2007-02-06

arxiv.org/abs/math/0702151v1

Mathematics

Algebraic Geometry

26 pages

Scientific paper

Let $X={\bf P}^2\times{\bf P}^{n-1}$ embedded with $\O(1,2)$. 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 L\"uroth 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 ${\bf P}^2$ by using the higher secant varieties of $X$.

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