Projective manifolds with small pluridegrees
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- by Mauro C. Beltrametti and Andrew J. Sommese PDF
- Trans. Amer. Math. Soc. 352 (2000), 3045-3064 Request permission
Abstract:
Let $\mathcal {L}$ be a very ample line bundle on a connected complex projective manifold $\mathcal {M}$ of dimension $n\ge 3$. Except for a short list of degenerate pairs $(\mathcal {M},\mathcal {L})$, $\kappa (K_\mathcal {M}+(n-2)\mathcal {L})=n$ and there exists a morphism $\pi : \mathcal {M} \to M$ expressing $\mathcal {M}$ as the blowup of a projective manifold $M$ at a finite set $B$, with $\mathcal {K}_M:=K_M+(n-2)L$ nef and big for the ample line bundle $L:= (\pi _*\mathcal {L})^{**}$. The projective geometry of $(\mathcal {M},\mathcal {L})$ is largely controlled by the pluridegrees $d_j:=L^{n-j}\cdot (K_M+(n-2)L)^j$ for $j=0,\ldots ,n$, of $(\mathcal {M},\mathcal {L})$. For example, $d_0+d_1=2g-2$, where $g$ is the genus of a curve section of $(\mathcal {M},\mathcal {L})$, and $d_2$ is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of $(\mathcal {M},\mathcal {L})$. In this article, a detailed analysis is made of the pluridegrees of $(\mathcal {M},\mathcal {L})$. The restrictions found are used to give a new lower bound for the dimension of the space of sections of $\mathcal {K}_M$. The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to $|2(K_\mathcal {M}+ (n-2)\mathcal {L})|$.References
- Mauro C. Beltrametti, M. Lucia Fania, and Andrew J. Sommese, On the adjunction-theoretic classification of projective varieties, Math. Ann. 290 (1991), no. 1, 31–62. MR 1107662, DOI 10.1007/BF01459237
- Mauro Beltrametti, Michael Schneider, and Andrew J. Sommese, Threefolds of degree $11$ in $\textbf {P}^5$, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 59–80. MR 1201375, DOI 10.1017/CBO9780511662652.006
- Mauro C. Beltrametti, Michael Schneider, and Andrew J. Sommese, Some special properties of the adjunction theory for $3$-folds in $\textbf {P}^5$, Mem. Amer. Math. Soc. 116 (1995), no. 554, viii+63. MR 1257080, DOI 10.1090/memo/0554
- Mauro C. Beltrametti and Andrew J. Sommese, Special results in adjunction theory in dimension four and five, Ark. Mat. 31 (1993), no. 2, 197–208. MR 1263551, DOI 10.1007/BF02559483
- Mauro C. Beltrametti and Andrew J. Sommese, On the adjunction-theoretic classification of polarized varieties, J. Reine Angew. Math. 427 (1992), 157–192. MR 1162435, DOI 10.1515/crll.1992.427.157
- M. C. Beltrametti and A. J. Sommese, On the dimension of the adjoint linear system for threefolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 1–24. MR 1315348
- Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR 1318687, DOI 10.1515/9783110871746
- Mauro C. Beltrametti and Andrew J. Sommese, On the second adjunction mapping. The case of a 1-dimensional image, Trans. Amer. Math. Soc. 349 (1997), no. 8, 3277–3302. MR 1401513, DOI 10.1090/S0002-9947-97-01809-6
- M.C. Beltrametti and A.J. Sommese, “On the degree and the birationality of the second adjunction mapping,” International Journal of Mathematics, to appear.
- Takao Fujita, Remarks on quasi-polarized varieties, Nagoya Math. J. 115 (1989), 105–123. MR 1018086, DOI 10.1017/S0027763000001562
- Eiji Horikawa, Algebraic surfaces of general type with small $C^{2}_{1}.$ I, Ann. of Math. (2) 104 (1976), no. 2, 357–387. MR 424831, DOI 10.2307/1971050
- A. Lanteri, M. Palleschi, and A. J. Sommese, On triple covers of $\mathbf P^n$ as very ample divisors, Classification of algebraic varieties (L’Aquila, 1992) Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 277–292. MR 1272704, DOI 10.1090/conm/162/01537
- Elvira Laura Livorni and Andrew John Sommese, Threefolds of nonnegative Kodaira dimension with sectional genus less than or equal to $15$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 4, 537–558. MR 880398
Additional Information
- Mauro C. Beltrametti
- Affiliation: Dipartimento di Matematica, Università Degli Studi di Genova, Via Dodecaneso 35, I-16146 Genova, Italy
- Email: beltrame@dima.unige.it
- Andrew J. Sommese
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- Email: sommese@nd.edu
- Received by editor(s): February 8, 1998
- Published electronically: May 21, 1999
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 3045-3064
- MSC (1991): Primary 14J40; Secondary 14M99, 14C20
- DOI: https://doi.org/10.1090/S0002-9947-99-02429-0
- MathSciNet review: 1641087