Degree of irrationalities of algebraic surfaces with Kodaira dimension zero

小平维数为零的代数曲面的无理度

• 批准号: 10640013
• 负责人: YOSHIHARA Hisao
• 金额: $ 1.73万
• 依托单位: NIIGATA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640013/
• 关键词: degree of irrationality; algebraic surface; Kodaira dimension; algebraic curve; Galois point; maximal rational intermediate subfield; 非有理次数; 極大有理中間体

1. Let Si (I = 1,2) be smooth projective surfaces and f : SィイD21ィエD2 → SィイD22ィエD2 be the surjective morphism. There has been a problem that whether the inequality dr(SィイD21ィエD2) 【greater than or equal】 dr(SィイD22ィエD2) hold true, where dr is the degree of irrationality. We have found examples which do not satisfy the inequality in the class of hyperelliptic surfaces.2. Let S be a hyperelliptic surface. We have proved that dr(S) = 2, 3 or 4, and that dr(S) = 2 if and only if 2KィイD2sィエD2 is trivial, where KィイD2sィエD2 is the canonical bundle of S.3. Let C be a plane curve of degree d and K = k(C) be the rational function field of C. We consider a projection пp of C from P ∈ C to a line l =ィイD4〜ィエD4 PィイD11ィエD1. The projection induces the extension of fields πィイD2PィエD2ィイD1*ィエD1 : k(l) * K. We have studied the structure of this extension from geometrical viewpoint. If the extension is Galois, we call P Galois point. We have determined all the Galois points. If P is not a Galois point, we cosider the minimal splitting field Lp of the extension K/k(l) and the Galois group Gal(Lp/k(l)). Our study have been done in this latter case only for d = 4, the general case must be studied in the furture.

1。令Si（i = 1,2）为光滑的射射表面，F：SYI D21E D2→SYI D22E D2是过滤的形态。存在一个问题，即不平等DR（SYI D21E D2）[大于或相等] DR（SYI D22E D2）是正确的，其中DR是非理性的程度。我们发现了无法满足过椭圆形表面类别的不平等的例子。2。令S为虚拟表面。我们已经证明了DR（s）= 2、3或4，并且当2KII D2是微不足道时，DR（S）= 2，其中KII D2是S.3的规范束。令C为D度D和K = K（C）的平面曲线为C的合理函数场。我们考虑了从P∈C到线L = II D4到P 11 D1的C的投影9p。投影诱导场的扩展πd2p 1 d1 * d1：k（l） * k。我们从几何观点研究了该扩展的结构。如果扩展是加洛伊斯，我们称为p galois点。我们已经确定了所有Galois点。如果P不是Galois点，我们将延伸k/k（L）和Galois组GAL（LP/K（L））的最小分裂场LP。我们的研究仅在以后的情况下仅针对d = 4进行，一般情况必须在furture中进行研究。

项目成果

Hisao Yoshihara: "Degree of irrationality of hyperelliptic surfaces"Algebra Colloquium. (in press).

Hisao Yoshihara：“超椭圆曲面的无理度”代数讨论会。

Shinichi Tajima: "Perturbed Lame equation and Buslav phase" Ukraine Journal of Mathematics. (in press).

Shinichi Tajima：“扰动拉梅方程和布斯拉夫相”乌克兰数学杂志。

Kei Miura: "Field theory for function fields of the quintic Fermat curves"Communications in Algebra. (印刷中).

Kei Miura：“五次费马曲线函数域的场论”代数通讯（正在出版）。

Hisao Yoshihara: "A note on the inequality of degrees of irrationalities of algebraic surfaces"Journal of Algebra. 207. 272-275 (1998)

Hisao Yoshihara：“关于代数曲面无理数度不等式的注释”代数杂志。

Hisao Yoshihara: "A note on the inequality of degrees of irrationalities of algebraic surface" Journal of Algebra. 207. 272-275 (1998)

Hisao Yoshihara：“关于代数曲面的无理数度不等式的注释”代数杂志。