Osaka Journal of Mathematics

Varieties of Picard rank one as components of ample divisors

Andrea Luigi Tironi

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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 $\mathcal{V}$.

Article information

Source
Osaka J. Math., Volume 52, Number 3 (2015), 601-617.

Dates
First available in Project Euclid: 17 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1437137611

Mathematical Reviews number (MathSciNet)
MR3370468

Zentralblatt MATH identifier
1331.14011

Subjects
Primary: 14C20: Divisors, linear systems, invertible sheaves 14C22: Picard groups
Secondary: 14J40: $n$-folds ($n > 4$) 14J45: Fano varieties

Citation

Tironi, Andrea Luigi. Varieties of Picard rank one as components of ample divisors. Osaka J. Math. 52 (2015), no. 3, 601--617. https://projecteuclid.org/euclid.ojm/1437137611


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