论文标题
在阿贝里亚自动形态群体中
On Abelian Automorphism Groups of Hypersurfaces
论文作者
论文摘要
给定整数$ d \ ge 3 $和$ n \ ge 3 $。令$ g $为一个有限的阿贝尔集团，在复杂的投影空间中忠实地忠于$ d $的高度表面上，$ \ mathbb {p}^{n-1} $。假设可以将$ g \ subset pgl（n，\ mathbb {c}）$提升为$ gl（n，\ mathbb {c}）$的子组。此外，假设存在$ g $中的元素$ g $，以便$ g/\ langle g \ rangle $具有订单coprime至$ d-1 $。然后确定所有可能的$ g $（定理4.3）。作为一个应用程序，我们得出（定理4.8）对于任何给定$（d，n）$的平滑性超曲面的线性自动形态的所有可能顺序。特别是，我们表明（命题5.1）表明，平滑立方四倍的自动形态的顺序为21、30、32、33、36或48，而这6个数字中的每一个都是通过独特的（达到同构）立方四倍来实现的。
Given integers $d\ge 3$ and $N\ge 3$. Let $G$ be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree $d$ in the complex projective space $\mathbb{P}^{N-1}$. Suppose $G\subset PGL(N, \mathbb{C})$ can be lifted to a subgroup of $GL(N,\mathbb{C})$. Suppose moreover that there exists an element $g$ in $G$ such that $G/\langle g\rangle$ has order coprime to $d-1$. Then all possible $G$ are determined (Theorem 4.3). As an application, we derive (Theorem 4.8) all possible orders of linear automorphisms of smooth hypersurfaces for any given $(d,N)$. In particular, we show (Proposition 5.1) that the order of an automorphism of a smooth cubic fourfold is a factor of 21, 30, 32, 33, 36 or 48, and each of those 6 numbers is achieved by a unique (up to isomorphism) cubic fourfold.
