论文标题
第二个基本形式和更高的高斯地图
Second fundamental form and higher Gaussian maps
论文作者
论文摘要
在本文中，我们显示了在平滑的投影曲线$ g \ \ geq 4 $的投影曲线与托雷利地图的第二个基本形式的较高的高斯图之间的关系。这概括了科伦坡，皮罗拉和托尔托拉在第二个高斯地图和第二个基本形式中获得的结果。结果，我们证明，对于任何非hyperelliptic曲线，高斯地图$μ__{6G-6} $都是注入性的，因此，所有$ k> 3G-3 $的Ghussian Maps $μ__{2K} $对于所有$ k> 3G-3 $都是零。我们还给出了$μ_{2k} $的等级，$ g-1 \ 1 \ leq k \ leq 3g-3。
In this paper we show a relation between higher even Gaussian maps of the canonical bundle on a smooth projective curve of genus $g \geq 4$ and the second fundamental form of the Torelli map. This generalises a result obtained by Colombo, Pirola and Tortora on the second Gaussian map and the second fundamental form. As a consequence, we prove that for any non-hyperelliptic curve, the Gaussian map $μ_{6g-6}$ is injective, hence all even Gaussian maps $μ_{2k}$ are identically zero for all $k >3g-3$. We also give an estimate for the rank of $μ_{2k}$ for $g-1 \leq k \leq 3g-3.$
