Semipolarized nonruled surfaces with sectional genus two

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 κ(S) = 0 and 2. Structure results for (S, L) are provided in both cases, according to the values of L2. When S is not minimal we describe explicitly the structure of any birational morphism from S to its minimal model S0, reducing the study of (S, L) to that of (S0, L 0), where L0 is a numerically effective line bundle with g(S0, L0) = 2 or 3. Our description of (S, L) when S is minimal, as well as that of the pair (S0, L0) when g(S0, L0) = 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