Osaka Journal of Mathematics

Varieties of Picard rank one as components of ample divisors

Andrea Luigi Tironi

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

https://projecteuclid.org/euclid.ojm/1437137611

Mathematical Reviews number (MathSciNet)
MR3370468

Zentralblatt MATH identifier
1331.14011

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

References

• 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.