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 k(S)=0 and 2. Structure results 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', reducing the study of (S,L) to that of (S',L'), where L' is a numerically effective line bundle with g(S',L')=2 or 3. Our description of (S,L) when S is minimal, as well as that of the pair (S',L') when g(S',L')=3 relies on on several results concerning linear systems, mainly on surfaces of Kodaira dimension zero. 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.