Symplectic bundles on the plane, secant varieties and L\"uroth quartics revisited

TL;DR

This paper studies secant varieties of a specific embedded product of projective spaces, revealing their hypersurface nature and connections to L"uroth quartics, with implications for symplectic bundles on the plane.

Contribution

It demonstrates that certain secant varieties are hypersurfaces and links these to classical algebraic geometry objects, providing new insights into symplectic bundles and secant varieties.

Findings

(n+1)-secant variety of X is a hypersurface.

Equation of the secant variety is a symmetric analog of Strassen's equation.

Connections established between secant varieties, L"uroth quartics, and symplectic bundles.

Abstract

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 $X$.