On quasi-polarized surfaces of general type whose sectional genus is equal to the irregularity
In the paper under review, the author continues his study of quasi-polarized surfaces \((X,L)\) over the complex number field, whose irregularity \(q(X)\) equals the sectional genus \(g(L)\). He begun this study in the papers [Can. J. Math. 50, No. 6, 1209--1235 (1998; Zbl 0930.14002) and Trans. Am. Math. Soc. 348, No. 10, 4185--4197 (1996; Zbl 0878.14005)], where he classified quasi-polarized surfaces with \(g(L)=q(X)\), such that either the Kodaira dimension is \(\leq 1\), or the Kodaira dimension is \(2\), \(L\) is ample and \(h^0(L)>0\). Here he assumes that the Kodaira dimension is \(2\) and \(h^0(L)>0\), but the hypothesis of ampleness of \(L\) is removed: \(L\) is only assumed to be nef and big. The main theorem asserts that, in this situation, \(h^0(L)=1\) and that, called \(D\) the effective divisor in \(| L|\), \(D\) is reduced and is of one of the following two types: a smooth irreducible curve with \(D^2\leq 4\), or a union of irreducible smooth curves \(C_i\), forming a configuration which is completely described. Several examples are given of the possible situations.
scientific article; zbMATH DE number 1270421