On the Picard group of Enriques surfaces
On the Picard group of Enriques surfaces
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在恩里克斯曲面的皮卡德群上
DOI：
10.1007/bf01456135
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发表时间：
1985
影响因子：
1.4
通讯作者：
F. Cossec
中科院分区：
文献类型：
--
作者：
F. Cossec
特征为4：2的代数闭域k上的Enriques曲面是非奇异射影曲面S，满足H ~（S，d~s）= H2（S，Cs）= 0和2Ks = 0.由Ks定义的可解二重覆盖是一个K3曲面R，它是一个非奇异射影曲面，满足Hi（R，d~R）=0，KR =0. Illusie在[1]中证明了模数值等价的因子群同构于Enriques格U ~ E 8（1），其中U和Es（1）分别表示惯性指数为（1，1）和（0，8）的唯一偶幺模格.本文的目的是利用这个同构来研究S的Picard群。证明了算术亏格为0或1的不可约曲线的某些构型的存在性，并由此推出了S和R的某些投射模型的存在性。例如，我们证明了以下结果：
An Enriques surface over an algebraically closed field k of characteristic 4:2 is a non-singular projective surface S with H ~ (S, d~s)= H2(S, Cs)= 0 and 2Ks = 0. The unramitied double cover defined by Ks is a K3 surface R, a non-singular projective surface with Hi(R, d~R)=0 , KR =0. Illusie has shown, [I], that the group of divisors modulo numerical equivalence is isomorphic to the Enriques lattice U ~ E 8 ( 1) where U and E s ( 1) denote, respectively, the unique even unimodular lattices of index of inertia (1,1) and (0, 8). The purpose of this note is to use this isomorphism to study the Picard group of S. We prove the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce from them the existence of certain projective models for S and R. For example, we prove the following results: