Picard groups of Zariski surfaces. (English) Zbl 0624.14021

This article contains a complete proof of the results an outline of which the author gave in C. R. Acad. Sci., Paris, Sér. I 297, 299-302 (1983; Zbl 0573.14012). Let \(\tilde S\) be the minimal resolution of singularities of a generic Zariski surface \(\bar S,\) whose affine model is defined by the equation \(Z^ p-\sum_{0\leq i+j\leq p}T_{ij}X^ iY^ j=0\quad over\) an algebraically closed field of characteristic \(p\geq 5.\tilde S\) has only rational double points of type \(A_{p-1}\) as singularities. The author examines the structure of the Picard group of \(\tilde S\) and proves that p \(Pic(\tilde S)\subset p Pic^{ob}\tilde S\), where \(Pic^{ob}(\tilde S)\) is a subgroup of \(Pic(\tilde S)\) generated by hyperplane sections and exceptional divisors.
[See also the following review.]
Reviewer: H.Maeda


14C22 Picard groups
14J25 Special surfaces
Full Text: Numdam EuDML


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