Semipolarized nonruled surfaces with sectional genus two

Aldo Biancofiore, Maria Lucia Fania and Antonio Lanteri

Abstract: 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 $1$.

Classification (MSC2000): 14C20, 14J28, 14J29; 14J25

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