Varieties of Picard rank one as components of ample divisors

Abstract

Let $\mathcal{V}$ be an integral normal complex projective variety of dimension $n\ge 3$ and denote by $\mathcal{L}$ an ample line bundle on $\mathcal{V}$. By imposing that the linear system $|\mathcal{L}|$ contains an element $A=A_1+...+A_r,r\ge 1$, where all the $A_i$'s are distinct effective Cartier divisors with Pic$(A_i)=\mathbb{Z}$, we show that such a $\mathcal{V}$ is as special as the components $A_i$ of $A\in |\mathcal{L}|$. 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 Pic$(A_j)=\mathbb{Z}$ for $j=2,...,r,$ 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 $\mathcal{V}$.

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