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

Let X = P × P embedded with O(1, 2). We prove that its (n + 1)-secant variety σn+1(X) is a hypersurface, while it is expected that it fills the ambient space. The equation of σn+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 P by using the higher secant varieties of X . AMS Subject Classification. 14J60, 14F05, 14H50, 14N15, 15A72