Reducible hyperplane sections of 4-folds: semi-nef decompositions with low sectional genera

Let $X$ be a smooth complex variety of dimension $n\geq 3$ and $L$ an ample and spanned line bundle on $X$. Suppose that a divisor $D\in |L|$ is a sum of two irreducible smooth components $A_1$ and $A_2$, which intersect transversally. The decomposition $D=A_1+A_2$ is called semi-nef if at least one of the divisors $A_i$ is either numerically effective or it is an exceptional divisor of the first reduction of the pair $(X,L)$ in terms of the adjunction theory (that is, $A_i=P^{n-1}$, ${A_i}_{|A_i}={\scr O}(-1)$, $L_{|A_i}={\scr O}(1)$, ${K_X}_{|A_i}={\scr O}(-n+1)$). The paper under review provides a classification of such pairs $(X,L)$ for $n=4$ under the assumption that the sectional genus of each of these components is at most 1, that is $$g(A_i,L_{|A_i})=[(K_{A_i}+(n-2)L_{|A_i}) ·L^{n-2}_{|A_i}+2]/2\leq 1$$ for $i=1, 2$.