Projective manifolds with small pluridegrees

Authors:
Mauro C. Beltrametti and Andrew J. Sommese

Journal:
Trans. Amer. Math. Soc. **352** (2000), 3045-3064

MSC (1991):
Primary 14J40; Secondary 14M99, 14C20

Published electronically:
May 21, 1999

MathSciNet review:
1641087

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Abstract: Let be a very ample line bundle on a connected complex projective manifold of dimension . Except for a short list of degenerate pairs , and there exists a morphism expressing as the blowup of a projective manifold at a finite set , with nef and big for the ample line bundle . The projective geometry of is largely controlled by the pluridegrees for , of . For example, , where is the genus of a curve section of , and is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of . In this article, a detailed analysis is made of the pluridegrees of . The restrictions found are used to give a new lower bound for the dimension of the space of sections of . 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 .

**1.**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**, 10.1007/BF01459237**2.**Mauro Beltrametti, Michael Schneider, and Andrew J. Sommese,*Threefolds of degree 11 in 𝑃⁵*, 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**, 10.1017/CBO9780511662652.006**3.**Mauro C. Beltrametti, Michael Schneider, and Andrew J. Sommese,*Some special properties of the adjunction theory for 3-folds in 𝑃⁵*, Mem. Amer. Math. Soc.**116**(1995), no. 554, viii+63. MR**1257080**, 10.1090/memo/0554**4.**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**, 10.1007/BF02559483**5.**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**, 10.1515/crll.1992.427.157**6.**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****7.**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****8.**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**, 10.1090/S0002-9947-97-01809-6**9.**M.C. Beltrametti and A.J. Sommese, ``On the degree and the birationality of the second adjunction mapping,'' International Journal of Mathematics, to appear.**10.**Takao Fujita,*Remarks on quasi-polarized varieties*, Nagoya Math. J.**115**(1989), 105–123. MR**1018086****11.**Eiji Horikawa,*Algebraic surfaces of general type with small 𝐶²₁. I*, Ann. of Math. (2)**104**(1976), no. 2, 357–387. MR**0424831****12.**A. Lanteri, M. Palleschi, and A. J. Sommese,*On triple covers of 𝐏ⁿ 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**, 10.1090/conm/162/01537**13.**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**

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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

DOI:
http://dx.doi.org/10.1090/S0002-9947-99-02429-0

Keywords:
Smooth complex polarized $n$-fold,
very ample line bundle,
adjunction theory,
log-general type,
pluridegrees.

Received by editor(s):
February 8, 1998

Published electronically:
May 21, 1999

Article copyright:
© Copyright 2000
American Mathematical Society