# zbMATH — the first resource for mathematics

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 $$1$$.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14J28 $$K3$$ surfaces and Enriques surfaces 14J29 Surfaces of general type 14J25 Special surfaces
Full Text: