论文标题
关于射影空间的平滑曲面的自动形态群
On the liftability of the automorphism group of smooth hypersurfaces of the projective space
论文作者
论文摘要
令$ x $为尺寸的平滑超出表面$ n \ geq 1 $和度数$ d \ geq 3 $在投影空间中，作为均匀形式$ f $的零集。如果$（n，d）\ neq（1,3），（2,4）$，众所周知，每种$ x $的自动形态均扩展到投射空间的自动形态，即$ \ operatoTorname {aut}（aut}（aut}（aut}（x）\ subseteq \ subSeteeq \ subsetOrtatOrnOrName {pgl} {pgl} $ {我们说，如果存在$ \ operatoratorname {gl}（n+2，\ m athbb {c}）$ projecting iSomorthical将$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ perepatotorname {aut of fromplightiant。本文我们的主要结果表明，每一个平滑的尺寸$ n $ and $ d $ $ f $ $ f $的自动形态组可在且仅当$ d $和$ d $和$ n+2 $相对较好时。我们还提供了一个有效的标准，以计算所有是质数的力量的整数，并且看起来像是尺寸平稳的超出表面$ n $和度量$ d $的自动形态的顺序。作为一个应用程序，我们提供了足够的条件，其中一些$ \ operatatorName {aut}（x）$的Sylow $ p $ -subgroup是订单$ p $的微不足道或循环。
Let $X$ be a smooth hypersurface of dimension $n\geq 1$ and degree $d\geq 3$ in the projective space given as the zero set of a homogeneous form $F$. If $(n,d)\neq (1,3), (2,4)$ it is well known that every automorphism of $X$ extends to an automorphism of the projective space, i.e., $\operatorname{Aut}(X)\subseteq \operatorname{PGL}(n+2,\mathbb{C})$. We say that the automorphism group $\operatorname{Aut}(X)$ is $F$-liftable if there exists a subgroup of $\operatorname{GL}(n+2,\mathbb{C})$ projecting isomorphically onto $\operatorname{Aut}(X)$ and leaving $F$ invariant. Our main result in this paper shows that the automorphism group of every smooth hypersurface of dimension $n$ and degree $d$ is $F$-liftable if and only if $d$ and $n+2$ are relatively prime. We also provide an effective criterion to compute all the integers which are a power of a prime number and that appear as the order of an automorphism of a smooth hypersurface of dimension $n$ and degree $d$. As an application, we give a sufficient condition under which some Sylow $p$-subgroups of $\operatorname{Aut}(X)$ are trivial or cyclic of order $p$.
