On polarized manifolds of \(\Delta\) -genus two. I (Q800433)

Given an ample line bundle L on a projective manifold M, the \(\Delta\) - genus of the polarized manifold (M,L) is defined by \(\Delta (M,L)=n+L^ n-h^ 0(M,L),\) where \(n=\dim M\). Polarized manifolds with \(\Delta\leq 1\) were classified by the author. This article is devoted to the case \(\Delta =2\). In {\S} 0 the classification is briefly outlined. The subsequent sections are devoted to the case in which \(\dim Bs| L| >0\) and \(L^ n>1\). It is first shown that \(| L|\) yields a certain fibration of M with fiber being curves. Thereafter the case of hyperelliptic fibration is precisely described and completely classified. Finally deformations of them are studied.