High dimensional reducible hyperplane sections with multigenera $\leq$ 1

Abstract.

Let X be an algebraic submanifold of the complex projective space $\mathbb{P}^N$ of dimension $n \geq 5$. We describe those $X \subset \mathbb{P}^N$ whose intersection with some hyperplane is a smooth simply normal crossing divisor $A_{1} + \cdots + A_{r}$ with $r \geq 2$ such that $g(A_{k}, L_{A_k}) \leq 1$ for $k=1,\ldots, r$.