Varieties of Picard rank one as components of ample divisors

Abstract

Let $\mathcal{V}$ be an integral normal complex projective variety of dimension $n \geq 3$ and denote by $\mathcal{L}$ an ample line bundle on $\mathcal{V}$. By imposing that the linear system $\lvert\mathcal{L}\rvert$ contains an element $A = A_{1} + \cdots + A_{r}$, $r \geq 1$, where all the $A_{i}$'s are distinct effective Cartier divisors with $\mathrm{Pic}(A_{i}) = \mathbb{Z}$, we show that such a $\mathcal{V}$ is as special as the components $A_{i}$ of $A \in \lvert\mathcal{L}\rvert$. After making a list of some consequences about the positivity of the components $A_{i}$, we characterize pairs $(\mathcal{V}, \mathcal{L})$ as above when either $A_{1} \cong \mathbb{P}^{n-1}$ and $\mathrm{Pic}(A_{j}) = \mathbb{Z}$ for $j = 2, \ldots, r$, or $\mathcal{V}$ is smooth and each $A_{i}$ is a variety of small degree with respect to $[H_{i}]_{A_{i}}$, where $[H_{i}]_{A_{i}}$ is the restriction to $A_{i}$ of a suitable line bundle $H_{i}$ on $\mathcal{V}$.