On the inseparability of the Gauss map
On the inseparability of the Gauss map
复制标题
论高斯图的不可分性
DOI：
10.1090/conm/123/1143550
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发表时间：
1990
影响因子：
0.6
通讯作者：
R. Piene
中科院分区：
文献类型：
--
作者：
S. Kleiman;R. Piene
设X为至少为2次的光滑完全交，并考虑以下两个新猜想：(1)一般嵌入切空间在唯一点上相切；换句话说，高斯映射是完全不可分割的。(2)如果X是至少3次的超曲面，其对偶超曲面也是光滑的，则X要么是具有特征2的平面三次曲面，要么是方程L: xj+1 = 0的费马超曲面，其中q是特征的幂次。如果X是一条曲线，这两个猜想都成立；本文在X是曲面时证明了这两个定理，在X是n折n ~ 3时证明了这两个定理。此外，任意投影曲线的高斯映射的可分离度与顶点数有关。
Let X be a smooth complete intersection of degree at least 2, and consider these two new conjectures: (1) A general embedded tangent space is tangent at a unique point; in other words, the Gauss map is purely inseparable. (2) If X is a hypersurface of degree at least 3 whose dual hypersurface is also smooth, then X is either a plane cubic in characteristic 2 or the Fermat hypersurface with equation L: xj+1 = 0 where q is a power of the characteristic. Both conjectures are known to hold if X is a curve; in this paper, they are proved if X is a surface, and supported if X is an n-fold, n ~ 3. In addition, the separable degree of the Gauss map of an arbitrary projective curve Cis related to the number of cusps.