High dimensional reducible hyperplane sections with multigenera $leq$ 1

摘要

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 $k=1,ldots, r$.