Osaka Journal of Mathematics

Varieties of Picard rank one as components of ample divisors

Andrea Luigi Tironi

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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

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

First available in Project Euclid: 17 July 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Tironi, Andrea Luigi. Varieties of Picard rank one as components of ample divisors. Osaka J. Math. 52 (2015), no. 3, 601--617.

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  • L. Bădescu: On ample divisors, Nagoya Math. J. 86 (1982), 155–171.
  • M.C. Beltrametti and A.J. Sommese: The Adjunction Theory of Complex Projective Varieties, de Gruyter Expositions in Mathematics 16, de Gruyter, Berlin, 1995.
  • K.A. Chandler, A. Howard and A.J. Sommese: Reducible hyperplane sections, I, J. Math. Soc. Japan 51 (1999), 887–910.
  • D. Eisenbud: Commutative Algebra, Graduate Texts in Mathematics 150, Springer, New York, 1995.
  • \begingroup T. Fujita: Vanishing theorems for semipositive line bundles; in Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer, Berlin, 1983, 519–528. \endgroup
  • T. Fujita: Classification Theories of Polarized Varieties, London Mathematical Society Lecture Note Series 155, Cambridge Univ. Press, Cambridge, 1990.
  • R. Hartshorne: Ample Subvarieties of Algebraic Varieties, Lecture Notes in Mathematics 156, Springer, Berlin, 1970.
  • R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
  • P. Ionescu: Embedded projective varieties of small invariants; in Algebraic Geometry, Bucharest 1982 (Bucharest, 1982), Lecture Notes in Math., 1056, Springer, Berlin, 1984, 142–186.
  • P. Ionescu: Generalized adjunction and applications, Math. Proc. Cambridge Philos. Soc. 99 (1986), 457–472.
  • P. Ionescu: On manifolds of small degree, Comment. Math. Helv. 83 (2008), 927–940.
  • R. Lazarsfeld: Positivity in Algebraic Geometry, I, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge, A Series of Modern Surveys in Mathematics 48, Springer, Berlin, 2004.
  • R. Lazarsfeld: Positivity in Algebraic Geometry, II, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge, A Series of Modern Surveys in Mathematics 49, Springer, Berlin, 2004.
  • A.J. Sommese: On manifolds that cannot be ample divisors, Math. Ann. 221 (1976), 55–72.
  • A.J. Sommese: Submanifolds of Abelian varieties, Math. Ann. 233 (1978), 229–256.
  • A.L. Tironi: Ample normal crossing divisors consisting of two del Pezzo manifolds, Forum Math. 22 (2010), 667–682.