Symplectic bundles on the plane, secant varieties and L"uroth quartics revisited
Symplectic bundles on the plane, secant varieties and L"uroth quartics revisited
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平面辛丛、割线簇和 L"uroth 四次重温
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发表时间：
2007
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通讯作者：
G. Ottaviani
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作者：
G. Ottaviani
$X={f P}^2 艾美斯f P}^{n-1}$嵌入$O（1，2）$。我们证明了它的$（n+1）$-割线簇$sigma_{n+1}（X）$是一个超曲面，同时期望它充满周围空间。σ_{n+1}（X）方程是斯特拉森方程的对称模拟。当n=4时，行列式映射把σ_5（X）带到L“uroth四次曲面的超曲面上，这是LePotier和Tikhomirov研究的Barth映射的映象.利用这一线索，得到了${f P}^2$通过使用$X$的高割线变种。
Let $X={f P}^2 imes{f P}^{n-1}$ 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 ${f P}^2$ by using the higher secant varieties of $X$.