Open algebraic surfaces of logarithmic Kodaira dimension one (Q2850559)

Let \(X\) be a smooth projective surface defined over an algebraically closed field of characteristic \(p\geq 0\) and let \(D\) be a reduced effective divisor on \(X\) having simple normal crossings. Assume the logarithmic Kodaira dimension \(\kappa(X-D)\) is \(1\).NEWLINENEWLINEIn case \(p=0\), by a result of \textit{Y. Kawamata} [Algebraic geometry, Proc. Summer Meet., Copenh. 1978, Lect. Notes Math. 732, 215--232 (1979; Zbl 0407.14012)] for some positive integer \(n\) the linear system \(|(K_X+D)^+|\) (where `\(+\)' denotes taking the positive part of the Zariski decomposition) defines a fibration over a curve whose general fiber is irreducible of arithmetic genus \(0\) or \(1\). Analysis of the fibration is the source of our understanding of \((X,D)\). By contracting some \((-1)\)-curves in \(X\) behaving nicely with respect to \(D\) (see the `theory of peeling', \S 2.3 in [\textit{M. Miyanishi}, Open algebraic surfaces. CRM Monograph Series. 12. Providence, RI: American Mathematical Society (AMS). (2001; Zbl 0964.14030)] we may assume that \((X,D)\) is almost minimal, i.e., that the morphism onto the log minimal model \((X,D)\to (\bar X,\bar D)\), for which \(K_{\bar X}+\bar D\) is numerically effective, contracts only curves in \(D\). Then the detailed description of the fibration is given in \S 2.6.3 [loc. cit.]NEWLINENEWLINEFollowing and modifying the arguments of Kawamata and Miyanishi, the author proves the existence and gives an analogous description of the above fibration in case \(\text{char} k>0\) (Theorem 2.1). Using it he proves that if \(S\) is a normal affine surface whose smooth part \(S_0\) has \(\kappa(S_0)=1\) then some concrete logarithmic \(m\)-genera \(\bar P_m(S_0)\) are nonzero. For example, in case \(S\) is smooth it is \(m=2\) and in case \(\text{char} k\neq 2\) it is \(m=4\) or \(m=6\).NEWLINENEWLINEFor the entire collection see [Zbl 1266.14004].