Varieties of Picard rank one as components of ample divisors

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}.