On smooth quartic embedding of Kummer surfaces
On smooth quartic embedding of Kummer surfaces
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Kummer 曲面的光滑四次嵌入
DOI：
10.3792/pjaa.67.223
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发表时间：
1991
期刊：
影响因子：
--
通讯作者：
Isao Naruki
中科院分区：
文献类型：
--
作者：
Isao Naruki
1. 本注释的目的是为了说明以下事实：对于任何允许用约化的pfaffian 3极化的阿贝尔曲面，我们总能构造相关Kummer曲面的双态射成P，(C)，其像是一个四次曲面。当阿贝尔曲面不能被主极化时，态射是光滑嵌入。我们还将围绕这个事实讨论一些几何问题。设A为阿贝尔曲面，E为A的全称覆盖，G为格，A=E/G。假设给a一个极化（充足的线束）H，我们用它的黎曼形式来识别H。（参见Weil b[4]。）因此，H是E上的厄米特形式，它的虚部在G上为z值，我们假设H的约简式为3，也就是说，虚部在G上的行列式等于9。我们用V表示类型为（2H, 1）（1：平凡半字符）的奇函数的空间。V是四维的。由于半特征是平凡的，V中的函数必然在a的两个扭转点处消失。将V作为F（a，H）的子空间，我们得到a到P(V*)_P(C)的一个有理映射。(V*是V的对偶空间。对于复向量空间W，我们用P（W)表示射影空间(Wk{O})/Cx。）设S为与A相关的Kummer曲面，即A的商通过z0—z对合得到的最小解形式化。由于V中的所有元素都是奇数，因此有理映射将S映射到P(V*)，我们看到这个映射实际上是一个双分态射，并且映像是一个四次曲面。线束2H引出S上的线束，我们用（2H）表示。这是自交12，并且正交于例外除数E， （i= 1,2，…）S.线束
1. The purpose of the note is to show the following fact: For any abelian surface admitting a polarization with the reduced pfaffian three, one can always construct a birational morphism of the associated Kummer surface into P,(C) whose image is a quartic surface. The morphism is a smooth embedding if the abelian surface can not be principally polarized. We will also discuss some geometry around this fact. Let A be an abelian surface, E the universal cover of A and G the lattice such A=E/G. Suppose that a polarization (ample line bundle) H is given to A. We identify H with its Riemann form. (See Weil [4].) H is thus a hermitian form on E whose imaginary part is Z-valued over G. We assume that the reduced pfaffian of H is three, that is, that the determinant of the imaginary part over G is equal to nine. We denote the space of odd theta functions of type (2H, 1) (1: the trivial semi-character) by V. V is then fourdimensional. Since the semi-character is trivial, the theta functions in V necessarily vanish at the two-torsion points of A. Regarding V as a subspace of F(A,H) we obtain a rational mapping of A into P(V*)_P(C). (V* is the dual space of V. For a complex vector space W we denote by P(W) the projective space (Wk{O})/Cx.) Let S be the Kummer surface associated with A i.e. the minimal desingularizatioa of the quotient of A by the involution z o--z. Since all elements in V are odd, the rational mapping induces a rational mapping of S into P(V*) and we see that this mapping is actually a birational morphism and that the image is a quartic surface. The line bundle 2H induces a line bundle over S, which we denote by (2H). This is of self intersection twelve and is orthogonal to the exceptional divisors E, (i=1, 2, ..., 16) of the desingularization S. The line bundle