On certain elliptic surfaces with maximal picard number
On certain elliptic surfaces with maximal picard number
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在某些具有最大皮卡数的椭圆面上
DOI：
10.1016/0040-9383(85)90054-0
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发表时间：
1985
期刊：
影响因子：
--
通讯作者：
Mangala Nori
中科院分区：
文献类型：
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作者：
Mangala Nori
所谓曲面，我们总是指定义在C上的光滑射影曲面。众所周知，对于基B上的椭圆曲面X，Picard数N由公式N= r+ 2+~（mv-1）给出，其中r是在B的类属点上亏格为1的v= 1的伴随曲线的除子群的秩，my表示奇异纤维C~中不可约分支的数目，{Cv} 1< v<，是椭圆曲面的奇异纤维。当椭圆曲面X在基B上有一个截面时，r实际上是相关椭圆曲线在B的一般点上的Mordell-Weil群的秩。由于Picard数定义为dimoHl（X，f~ lx）c~ H2（X，Q），因此很清楚N<hL 1，其中hLI = dimcHI（X，~）.本文研究h1 '= N，r= 0的椭圆曲面.我们证明了以下定理，它分类的情况下，J（X的功能不变量在科代拉）是非常数的曲面。J的定义见§ 1。对于pe B，用ep表示J在p处的分歧指数。
BY A SURVACE we shall always mean a smooth projective surface defined over C. It is well-known that for an elliptic surface X over a base B the Picard Number N is given by the formula N= r+ 2+~(mv-1) where r is the rank of the divisor class group of the v= l associated curve of genus one over the generic point of B and my denotes the number of irreducible components in the singular fiber C~,{Cv} 1< v<, being the singular fibers of the elliptic surface. When the elliptic surface X has a section over the base B, r is in fact the rank of the Mordell-Weil group of the associated elliptic curve over the generic point of B. Since the Picard Number is defined as dimoH l (X, f~ lx) c~ H 2 (X, Q) it is clear that N< h L1 where h LI= dim cHI (X,~).The purpose of this paper is to study elliptic surfaces for which h 1'= N and r= 0. We prove the following theorem which classifies such surfaces in the case J (the functional invariant of X in the sense of Kodaira) is nonconstant. See § 1 for definition of J. Denote the ramification index of J at p by ep for pe B.