论文标题
椭圆表面上代数品种和矢量场的无穷小型自动形态
Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces
论文作者
论文摘要
我们给出了几个有关连接的组件$ {\ rm aut} _x^0 $在一个字段上适当品种$ x $的自动形态方案的$，例如其在birational修改，归一化，限制封闭子理学和变形方面的限制和变形。然后，我们将结果应用于研究雅各布椭圆形表面$ f：x \ to c $在代数封闭的田地上的自动形态方案，从而使鲁达科夫和沙法雷维奇的工作推广，同时对他们的某些陈述进行了反示例。如果$ f $的通用纤维是普通的，或者如果$ f $不承认，并且没有这些假设，那么$ f $的通用纤维是普通的，或者如果没有这些假设，对于任何基本culve $ c $ c $ c $ c $ c $ h^0（x，t_x），我们可以在任何假设上，对于任何基本cureve $ c $ c $ c $ c $ h^0（x，t_x），将限制全球矢量场$ h^0（x，t_x）$的尺寸$ h^0（x，t_x）$。如果$ f $不是各向同性的，我们证明$ {\ rm aut} _x^0 \congμ__{p^n} $，并以$ c $的属和多个$ f $的多纤维的属性对$ n $进行了限制。作为推论，我们重新启动了在K3表面上全球向量场的不存在，并计算了具有特征性$ 2 $的通用超高式Enriques表面的自动形态方案的连接组件。最后，我们在椭圆表面上的水平和垂直组方案动作提出了其他结果，这些结果可用于确定$ {\ rm aut} _x^0 $在许多具体情况下明确地。
We give several results concerning the connected component ${\rm Aut}_X^0$ of the automorphism scheme of a proper variety $X$ over a field, such as its behaviour with respect to birational modifications, normalization, restrictions to closed subschemes and deformations. Then, we apply our results to study the automorphism scheme of not necessarily Jacobian elliptic surfaces $f: X \to C$ over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension $h^0(X,T_X)$ of the space of global vector fields on an elliptic surface $X$ if the generic fiber of $f$ is ordinary or if $f$ admits no multiple fibers, and show that, without these assumptions, the number $h^0(X,T_X)$ can be arbitrarily large for any base curve $C$ and any field of positive characteristic. If $f$ is not isotrivial, we prove that ${\rm Aut}_X^0 \cong μ_{p^n}$ and give a bound on $n$ in terms of the genus of $C$ and the multiplicity of multiple fibers of $f$. As a corollary, we re-prove the non-existence of global vector fields on K3 surfaces and calculate the connected component of the automorphism scheme of a generic supersingular Enriques surface in characteristic $2$. Finally, we present additional results on horizontal and vertical group scheme actions on elliptic surfaces which can be applied to determine ${\rm Aut}_X^0$ explicitly in many concrete cases.
