×

On the Picard number of a Fermat surface. (English) Zbl 0567.14021

A general algorithm is considered for the computation of the Picard number of the Fermat surface \(X^ 2_ m=\{x^ m_ 0+x_ 1^ m+x_ 2^ m+x_ 3^ m=0\}.\) If m is not prime to 2 and 3 then the Picard number is \(\rho (X^ 2_ m)=3(m-1)(m-2)+1;\) moreover, \(NS(X^ 2_ m)\otimes {\mathbb{Q}}\) is generated by classes of straight lines on \(X^ 2_ m\). The values \(\rho (X^ 2_ m)\) have been computed also for \(m=2^ n\) and \(m=3^ n\). At the end of the article it is noted that Aoki has recently obtained a general formula for \(\rho (D^ 2_ m)\) [cf. N. Aoki, Math. Ann. 266, 23-54 (1983; Zbl 0506.14030) and ibid. 267, 572 (1984; Zbl 0534.14021)].

MSC:

14C22 Picard groups
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
14J25 Special surfaces
PDFBibTeX XMLCite