Reducible hyperplane sections of threefolds: two components of sectional genus zero
Let \(X\) be a smooth complex projective variety, \(L\) a (very) ample divisor on \(X\) and \(A\) an element in the linear system \(| L| \). There are many results describing the variety \(X\) under special assumptions on \(A\); for a general account on this problem see for instance [\textit{M. C. Beltrametti} and \textit{A. J. Sommese}, The adjunction theory of complex projective varieties. De Gruyter Expositions in Mathematics. 16. (Berlin: de Gruyter) (1995; Zbl 0845.14003)]. Most of these results assumed \(A\) to be smooth, or with mild singularities. Recently \textit{K. A. Chandler, A. Howard} and \textit{A. J. Sommese} [J. Math. Soc. Japan 51, No.4, 887--910 (1999; Zbl 0973.14018)] started the study of a more special situation, assuming \(A\) to be reducible, with smooth components \(A_i\) meeting with simple normal crossings, with some additional assumptions on the genera of the curve sections of the \(A_i\). The paper under review takes its place in this context, studying the case in which \(\dim X=3\) and \(L\) is an ample and spanned divisor such that \(| L| \) contains an element \(A+B\), where \(A\) and \(B\) are two smooth surfaces, one of which is nef, meeting transversally and of sectional genus \(g(A,L_A)=g(B,L_B)=0\). The assumption on the sectional genus yields that \(A\) and \(B\) are projective planes, two dimensional quadrics or rational ruled surfaces, with a suitable polarization. Depending on the cases, the authors show that either \(g(X,L) \leq 1\) or that \(K_X+2L\) is not ample (actually the nefness of one surface is used only in the case in which both \(A\) and \(B\) are rational ruled surfaces) and then they use results of adjunction theory to describe the pairs \((X,L)\).
scientific article; zbMATH DE number 2157862