On the second adjunction mapping. The case of a 1-dimensional image

Let $\widehat {L}$ be a very ample line bundle on an $n$-dimensional projective manifold $\widehat {X}$, i.e., assume that $\widehat {L}\approx i^*{\mathcal O}_{{\mathbb P}^{N}}(1)$ for some embedding $i:\widehat {X}\hookrightarrow {\mathbb P}^{N}$. In this article, a study is made of the meromorphic map, $\widehat {\varphi } : \widehat {X}\to \Sigma$, associated to $|K_{\widehat {X}}+(n-2)\widehat {L}|$ in the case when the Kodaira dimension of $K_{\widehat {X}}+(n-2)\widehat {L}$ is $\ge 3$, and $\widehat {\varphi }$ has a $1$-dimensional image. Assume for simplicity that $n=3$. The first main result of the paper shows that $\widehat \varphi$ is a morphism if either $h^0(K_{\widehat X}+\widehat L)\geq 7$ or $\kappa (\widehat {X})\geq 0$. The second main result of this paper shows that if $\kappa (\widehat X)\ge 0$, then the genus, $g(f)$, of a fiber, $f$, of the map induced by $\widehat \varphi$ on hyperplane sections is $\leq 6$. Moreover, if $h^0(K_{\widehat X}+\widehat L)\ge 21$ then $g(f)\leq 5$, a connected component $F$ of a general fiber of $\widehat \varphi$ is either a $K3$ surface or the blowing up at one point of a $K3$ surface, and $h^1({\mathcal O}_{\widehat X})\le 1$. Finally the structure of the finite to one part of the Remmert-Stein factorization of $\widehat \varphi$ is worked out.