This paper is concerned with exhibiting complex algebraic surfaces of general type having prescribed Picard number. For a complex algebraic surface \(X\), the Picard number satisfies the Lefschetz bound \(\rho(X)\leq h^{1,1}(X)\).
The paper focuses on quintic surfaces \(X\) in \({\mathbb{P}}^3\), where the Hodge numbers are given by \(h^{2,0}(X)=4=h^{0,2}(X)\) and \(h^{1,1}(X)=45\). The first question dealt with in this article is to realize quintic surface \(X\) whose Picard number attains the maximum value, namely \(45\).
Theorem 1. The surface \(Y\subset{\mathbb{P}}^3\) defined by the equation \[ yzw^3+xyz^3+wxy^3+zwx^3=0 \] has exactly four \(A_9\) singularities at the points where three coordinates vanish simultaneously. It minimal resolution \(X\) has maximum Picard number \(\rho(X)=45\).
Three proofs are presented. The first proof exploits the fact that \(X\) is a Galois quotient of some Fermat surface. The second proof exhibit rational curves on \(X\) that generate the Néron-Severi group \(\mathrm{NS}(X)\) up to finite index, and the third proof uses the cyclic group of order \(15\) acting on \(X\) to show that the \({\mathbb{Q}}\)-transcendental cycles form a one-dimensional vector space over the cyclotomic field \({\mathbb{Q}}(\zeta_{15})\).
The method of the first proof realizes some given numbers as Picard numbers.
Theorem 2. If \(r=1,5,13\) or an odd integer between \(17\) and \(45\), there there exists a quintic surface \(X\) with \(\rho(X)=r\).
As a byproduct of the first and the second proofs, the zeta-function of \(X\) can be computed.