论文标题
过椭圆形和广义的过偏变变形
Deformations of hyperelliptic and generalized hyperelliptic polarized varieties
论文作者
论文摘要
在本文中，我们研究了dimension $ m $和sectional属$ g $的过两极化品种$（x，l）$的变形，以使$ | l | $引起的形态$φ$的图像$ y $很流畅。如果$ l^m <2g-2 $，则众所周知，通过aidkuntion和clifford的定理，$（x，l）$的任何变形都是过度ellelliptic的。因此，我们专注于$ l^m = 2g-2 $或$ l^m = 2g $。我们证明，如果$（x，l）$是fano-k3，那么，除非$ y $是一个超等号，否则$（x，l）$的所有变形又是过度ellelliptic（如果$ y $是一个超等级，则$ $ $φ$的一般变形是嵌入的）。这与高纤维化规范曲线和过纤维k3表面的情况形成鲜明对比。如果$ l^m = 2g $，那么我们证明，在大多数情况下，$φ$的一般变形是$ 1 $的有限形态。这提供了$ 2 $形态的有趣示例，这些形态可能会变形为$ 1 $的形态。我们将结果扩展到所谓的普通过度偏光狂，卡拉比YAU和一般类型的品种。这些问题的解决方案与代数品种$ y $上的双重结构的存在或不存在密切相关。我们也解决了这一问题。
In this article we study the deformations of hyperelliptic polarized varieties $(X,L)$ of dimension $m$ and sectional genus $g$ such that the image $Y$ of the morphism $φ$ induced by $|L|$ is smooth. If $L^m < 2g-2$, it is known that, by adjunction and the Clifford's theorem, any deformation of $(X,L)$ is hyperelliptic. Thus, we focus on when $L^m=2g-2$ or $L^m=2g$. We prove that, if $(X,L)$ is Fano-K3, then, except when $Y$ is a hyperquadric, all deformations of $(X,L)$ are again hyperelliptic (if $Y$ is a hyperquadric, the general deformation of $φ$ is an embedding). This contrasts with the situation of hyperelliptic canonical curves and hyperelliptic K3 surfaces. If $L^m=2g$, then we prove that, in most cases, a general deformation of $φ$ is a finite morphism of degree $1$. This provides interesting examples of degree $2$ morphisms that can be deformed to morphisms of degree $1$. We extend our results to so-called generalized hyperelliptic polarized Fano, Calabi-Yau and general type varieties. The solutions to these questions are closely intertwined with the existence or non existence of double structures on the algebraic varieties $Y$. We address this matter as well.
