Galois points on quartic surfaces
Galois points on quartic surfaces
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四次曲面上的伽罗瓦点
DOI：
10.2969/jmsj/05330731
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发表时间：
2001
影响因子：
0.7
通讯作者：
Hisao Yoshihara
中科院分区：
文献类型：
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作者：
Hisao Yoshihara
设S是射影空间中的光滑超曲面，考虑S从PeS到平面f/的投影.这个投影导致域k（S）/k（H）的扩张。点P称为伽罗瓦点，如果扩张是伽罗瓦。我们研究的四次曲面的结构集中在伽罗瓦点。我们将证明伽罗瓦点的个数是0、1、2、4或8，并证明伽罗瓦点的分布规律的存在性。L.导论.设k是特征为零的代数闭域。“我们把它作为我们讨论的基础领域。设S是三射影空间P3：Pt（k）中d次光滑超曲面，其中d > 4.设K k（^g）为S的有理函数域。子域K1被称为极大有理子域，如果它是有理的，即，一个纯粹的超越扩展k，并不包含在任何其他合理的子域。研究扩张Kf K^的结构似乎很有趣。如果我们知道它，我们就能够对K的所有子域进行分类。因为，根据Zorkiki-Castelnuovo定理，K的任何子域（不是k）。是理性的因此，研究K和K^之间存在什么场就足够了。设L是K f K* 的伽罗瓦闭包，则需要研究伽罗瓦群Gal（L I K^）的结构.因此，我们首先要做的研究是找出什么时候扩张是伽罗瓦（cf. t6]）。这里“当”的意思有点模糊，如果我们考虑下面的4的模型，它将变得清晰。对于每个点P e，S，设ftp：5. -> H是S从P到平面H的投影。这个有理映射导出了域Klk（H）的扩张.我们知道，Sis d-1或d-2的非理性程度（参见。[1]，[0]），因此k（H）是极大有理子域。显然，这个扩张的结构不依赖于I，而是依赖于4，所以我们写e Kp而不是k（H）。因此，上面的问题等价于说对于P ∈ S的哪一点，扩张KlKp变成伽罗瓦。按照上述方法进行的研究已经对4度和5度曲线进行了研究（参见。[6]，t71）。20 A0数学学科分类。小学14 J70;中学14127，14 J28。
Let ,S be a smooth hypersurface in the projective three space and consider a projection of S from P e S to a plane f/. This projection induces an extension of fields k(S)/k(H). The point P is called a Galois point if the extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is zero, one, two, four or eight and the existence of some rule of distribution of the Galois points. l. Introduction. Let k be an algebraically closed field of characteristic zero. 'We fix it as the ground field of our discussion. Let S be a smooth hypersurface of degree d in the projective three space P3 : Pt (k), where we assume that d > 4. Let K k(^g) be the rational function field of S. A subfield K^ is said to be a maximal rational subfield if it is rational, i.€., a purely transcendental extension of k, and is not contained in any other rational subfield. It seems interesting to study the structure of the extension Kf K^. If we know it, we will be able to classify of all the subfields of K. Because, by Zariski-Castelnuovo's theorem any subfield (which is not k) of K. is rational. So that it is sufficient to study what fields exist between K and K^. Let L be the Galois closure of K f K*, then we need to study the structure of the Galois group Gal(L I K^). For that reason, the study we have to do first is to find when the extension is Galois (cf. t6]). Here the meaning "when" is a little ambiguous, it will become clear if we consider the model of 4 as follows. For each point P e ,S, let ftp: 5... -> H be a proje:tion of S from P to a plane H. This rational map induces the extension of fields K lk(H). We know that the degree of irrationality of Sis d-1 or d-2(cf. [1], [0]), hence k(H) is a maximal rational subfield. Clearly the structure of this extension does not depend on I/, but on 4, so that we writ e Kp instead of k(H). Therefore, the above question is equivalent to say for which point P e S the extension K lKp becomes Galois. The study following the above method has been done for curves of degrees 4 and 5 (cf. [6], t7l). 20A0 Mathematics Subject Classification. Primary 14J70; Secondary 14127, 14J28.