Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 47, No. 1, pp. 175-193 (2006)
Semipolarized nonruled surfaces with sectional genus two
Aldo Biancofiore, Maria Lucia Fania and Antonio LanteriDipartimento di Matematica, Università degli Studi di L'Aquila, Via Vetoio Loc. Coppito, 67100 L'Aquila, Italy, e-mail: email@example.com e-mail: firstname.lastname@example.org; Dipartimento di Matematica ``F. Enriques'' Università, Via C. Saldini 50, I-20133 Milano, Italy, e-mail: email@example.com
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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Electronic version published on: 9 May 2006. This page was last modified: 4 Nov 2009.