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Semipolarized nonruled surfaces with sectional genus two. (English) Zbl 1092.14008
Summary: Complex projective nonruled surfaces \(S\) endowed with a numerically effective line bundle \(L\) of arithmetic genus \(g(S,L)=2\) are investigated. In view of existing results on elliptic surfaces we focus on surfaces of Kodaira dimension \(\kappa(S)=0\) and \(2\). Structure results for \((S,L)\) are provided in both cases, according to the values of \(L^2\). When \(S\) is not minimal we describe explicitly the structure of any birational morphism from \(S\) to its minimal model \(S_0\), reducing the study of \((S,L)\) to that of \((S_0,L_0)\), where \(L_0\) is a numerically effective line bundle with \(g(S_0,L_0)=2\) or \(3\). Our description of \((S,L)\) when \(S\) is minimal, as well as that of the pair \((S_0,L_0)\) when \(g(S_0,L_0)=3\), relies on several results concerning linear systems, mainly on surfaces of Kodaira dimension \(0\). Moreover, several examples are provided, especially to enlighten the case in which \(S\) is a minimal surface of general type, \((S,L)\) having Iitaka dimension \(1\).

14C20 Divisors, linear systems, invertible sheaves
14J28 \(K3\) surfaces and Enriques surfaces
14J29 Surfaces of general type
14J25 Special surfaces
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