On Characteristic Systems of Families of Surfaces with Ordinary Singularities in a Projective Space
On Characteristic Systems of Families of Surfaces with Ordinary Singularities in a Projective Space
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射影空间中普通奇点曲面族的特征系统
DOI：
10.2307/2373235
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发表时间：
1965
影响因子：
1.7
通讯作者：
K. Kodaira
中科院分区：
文献类型：
--
作者：
K. Kodaira
在文[5]中，我们证明了在三重环中具有常奇点的解析曲面族的特征系的完备性定理。在本文中，我们研究的应用定理的曲面与普通奇性的射影3-空间。该定理断言，如果曲面是半正则的，则在环境三重中具有普通奇点的极大解析曲面族的特征系统是完备的。在第2节中，我们证明了在射影三维空间中，半正则性与正则性一致，并且正则性等价于某些线性方程组的线性无关性（定理2）。我们重新表述了射影三维空间中具有常奇点的解析曲面族的特征系统的完备性定理（定理1）。在第四节中，我们证明了正则性的一些准则（定理5，6和7）。在射影三维空间中，具有常奇性的完备连续曲面系的特征系的完备性的证明问题是由0。Zagliki（见Zagliki [9]，第99页）。我们在第2节和第4节中的结果可以被认为是对这个问题的部分回答。在第5节中，我们考虑曲面的几个具体例子。首先，我们显示的援助的标准，某些经典的例子，表面是正规的。我们研究了二重曲线为非奇异完全交的曲面，发现它们除了某些特殊情况外都不是正则的，但它们构成具有完全特征系的极大族。显然，这一结果表明，半正则性的要求对具有普通奇点的曲面施加了很强的限制。最后，我们研究定理1的应用，规范曲面的不规则0，亏格4和7阶的研究较早的恩里克斯[2]和麦克斯韦[7]。
In our previous paper [5] we have proved a theorem of completeness of characteristic systems for analytic families of surfaces with ordinary singularities in ambient threefolds. In this paper we examine the application of the theorem to surfaces with ordinary singularities in a projective 3-space. The theorem asserts that the characteristic systems of a maximal analytic family of surfaces with ordinary singularities in an ambient threefold are complete if the surfaces are semi-regular. We show, in Section 2, that, in a projective 3-space, the semi-regularity coincides with the regularity and that the regularity is equivalent to the linear independence of certain simultaneous linear equations (Theorem 2). We reformulate the theorem of completeness of characteristic systems for analytic families of surfaces with ordinary singularities in a projective 3-space (Theorem 1). In Section 4 we prove some criteria of regularity (Theorems 5, 6 and 7). The problem of proving the completeness of the characteristic systems of complete continuous systems of surfaces with ordinary singularities in a projective 3-space has been proposed by 0. Zariski (see Zariski [9], p. 99). Our results in Sections 2 and 4 may be considered as partial answers to this problem. In Section 5 we consider several concrete examples of surfaces. First we show with the aid of the criteria that certain classical examples of surfaces are regular. We theii examine surfaces whose double curves are non-singular complete intersections and find that they are not regular except some special cases whereas they form maximal families with complete characteristic systems. Apparently this result indicates that the requirement of semi-regularity imposes a strong restriction on surfaces with ordinary singularities. Finally we examine the application of Theorem 1 to canonical surfaces of irregularity 0, of genus 4 and of order 7 studied earlier by Enriques [2] and by Maxwell [7].