Polarized surfaces of -genus

Authors:
Maria Lucia Fania and Elvira Laura Livorni

Journal:
Trans. Amer. Math. Soc. **328** (1991), 445-463

MSC:
Primary 14C20; Secondary 14D20, 14J25

DOI:
https://doi.org/10.1090/S0002-9947-1991-0992607-7

MathSciNet review:
992607

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a smooth, complex, algebraic, projective surface and let be an ample line bundle on it. Let denote the -genus of the pair . The purpose of this paper is to classify such pairs under the assumption that and the complete linear system contains a smooth curve. If and , Fujita has shown that is very ample and . If and , then and those pairs have been studied by Fujita and Beltrametti, Lanteri, and Palleschi. To study the remaining cases we have examined the two possibilities of being nef or not, for . In the cases in which is nef it turned out to be very useful to iterate the adjunction mapping for ample line bundles as it was done by Biancofiore and Livorni in the very ample case. If there are still open cases to solve in which completely different methods are needed.

**[BLP]**M. Beltrametti, A. Lanteri, and M. Palleschi,*Algebraic surfaces containing an ample divisor of arithmetic genus two*, Ark. Mat.**25**(1987), 189-210. MR**923406 (89e:14037)****[BL]**A. Biancofiore and E. L. Livorni,*On the iteration of the adjunction process in the study of rational surfaces*, Indiana Math. J.**36**(1987), 167-187. MR**876997 (88d:14022)****[BL]**-,*On the iteration of the adjunction process for surfaces of negative Kodaira dimension*, Manuscripta Math.**64**(1989), 35-54. MR**994380 (90c:14004)****[BL]**-,*Algebraic ruled surfaces of low sectional genus*, Ricerche Mat.**36**(1987), 17-32. MR**977050 (89m:14021)****[F]**T. Fujita,*On the structure of polarized varieties of*-*genus zero*, J. Fac. Sci. Univ. Tokyo**22**(1975), 103-115. MR**0369363 (51:5596)****[F]**-,*On polarized varieties of small*-*genera*, Tôhoku Math. J.**34**(1982), 319-341. MR**676113 (85e:14013)****[F]**-,*On hyperelliptic polarized varieties*, Tôhoku Math. J.**35**(1983), 1-44. MR**695657 (84h:14016)****[F]**-,*On polarized manifolds of*-*genus two*, J. Math. Soc. Japan**36**(1984), 709-730. MR**759426 (85m:14015)****[F]**-,*On polarized manifolds whose adjoint bundles are not semipositive*, Advanced Study in Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 167-178. MR**946238 (89d:14006)****[F]**-,*Classification of polarized manifolds of sectional genus two*, preprint.**[F]**-,*On classification of polarized manifolds by sectional genus*, preprint.**[F]**-,*Polarized manifolds of degree three and*-*genus two*, J. Math. Soc. Japan**41**(1989), 311-331. MR**984755 (90e:14037)****[F]**-,*On the structure of polarized manifolds with total deficiency one*. III, J. Math. Soc. Japan**36**(1984), 75-89. MR**723595 (85e:14061)****[H]**R. Hartshorne,*Algebraic geometry*, Graduate Texts in Math., no. 52, Springer, 1977. MR**0463157 (57:3116)****[I]**P. Ionescu,*Ample and very ample divisors on surfaces*, Rev. Roumaine Math. Pures Appl.**33**(1988), 349-358. MR**950131 (89f:14037)****[La]**A. Lanteri,*On polarized surfaces of*-*genus two*, Ann. Mat. Pura Appl. (4)**151**(1988), 317-329. MR**964517 (90a:14053)****[LP]**A. Lanteri and M. Palleschi,*About the adjunction process for polarized algebraic surfaces*, J. Reine Angew. Math.**352**(1984), 15-23. MR**758692 (86h:14028)****[Li]**E. L. Livorni,*Classification of algebraic surfaces with sectional genus less than or equal to six*. III:*Ruled surfaces with*, Math. Scand.**59**(1986), 9-29. MR**873485 (88d:14024)****[Li]**-,*Classification of algebraic surfaces with sectional genus less than or equal to six*. II:*Ruled surfaces with*, Canad. J. Math.**38**(1986), 1110-1121. MR**869716 (88d:14023)****[S]**A. J. Sommese,*Hyperplane sections of projective surfaces*I.*The adjunction mapping*, Duke Math. J.**46**(1979), 377-401. MR**534057 (82f:14033)****[S]**-,*On the birational theory of hyperplane sections of projective threefolds*, unpublished 1981 manuscript.**[S]**-,*On the adjunction theoretic structure of projective varieties*, Proc. Conf. Complex Analysis and Algebraic Geometry (Gottingen, 1985), (H. Grauert, ed.), Lecture Notes in Math., vol. 1194, Springer, 1986. MR**855885 (87m:14049)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14C20,
14D20,
14J25

Retrieve articles in all journals with MSC: 14C20, 14D20, 14J25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-0992607-7

Keywords:
Surfaces,
-genus,
sectional genus

Article copyright:
© Copyright 1991
American Mathematical Society