On the adjunction mapping of very ample vector bundles of corank one

Lanteri, Antonio; Palleschi, Marino; Sommese, Andrew

doi:10.1090/S0002-9947-03-03278-1

On the adjunction mapping of very ample vector bundles of corank one
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by Antonio Lanteri, Marino Palleschi and Andrew J. Sommese PDF

Trans. Amer. Math. Soc. 356 (2004), 2307-2324 Request permission

Abstract:

Let $\mathcal {E}$ be a very ample vector bundle of rank $n-1$ over a smooth complex projective variety $X$ of dimension $n\geq 3$. The structure of $(X,\mathcal {E})$ being known when $\kappa (K_{X} + \det \mathcal {E}) \leq 0$, we investigate the structure of the adjunction mapping when $0 < \kappa (K_{X} + \det \mathcal {E}) < n$.

References

M. Andreatta, E. Ballico, and J. Wiśniewski, Vector bundles and adjunction, Internat. J. Math. 3 (1992), no. 3, 331–340. MR 1163727, DOI 10.1142/S0129167X92000114
M. C. Beltrametti and A. J. Sommese, Comparing the classical and the adjunction-theoretic definition of scrolls, Geometry of complex projective varieties (Cetraro, 1990) Sem. Conf., vol. 9, Mediterranean, Rende, 1993, pp. 55–74. MR 1225588
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
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
F. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. USSR Isvestija 13 (1979), 499–555.
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Robin Hartshorne, Ample vector bundles on curves, Nagoya Math. J. 43 (1971), 73–89. MR 292847
Paltin Ionescu, Embedded projective varieties of small invariants, Algebraic geometry, Bucharest 1982 (Bucharest, 1982) Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984, pp. 142–186. MR 749942, DOI 10.1007/BFb0071773
Paltin Ionescu, Embedded projective varieties of small invariants. III, Algebraic geometry (L’Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 138–154. MR 1040557, DOI 10.1007/BFb0083339
Antonio Lanteri and Hidetoshi Maeda, Ample vector bundle characterizations of projective bundles and quadric fibrations over curves, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 247–259. MR 1463183
Antonio Lanteri, Hidetoshi Maeda, and Andrew J. Sommese, Ample and spanned vector bundles of minimal curve genus, Arch. Math. (Basel) 66 (1996), no. 2, 141–149. MR 1367156, DOI 10.1007/BF01273345
Antonio Lanteri and Marino Palleschi, About the adjunction process for polarized algebraic surfaces, J. Reine Angew. Math. 352 (1984), 15–23. MR 758692
Hidetoshi Maeda and Andrew J. Sommese, Very ample vector bundles of curve genus two, Arch. Math. (Basel) 79 (2002), no. 1, 74–80. MR 1923041, DOI 10.1007/s00013-002-8287-0
Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI 10.2307/2007050
Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
Thomas Peternell, MichałSzurek, and Jarosław A. Wiśniewski, Fano manifolds and vector bundles, Math. Ann. 294 (1992), no. 1, 151–165. MR 1180456, DOI 10.1007/BF01934319
Miles Reid, Bogomolov’s theorem $c_{1}^{2}\leq 4c_{2}$, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 623–642. MR 578877
Igor Reider, Vector bundles of rank $2$ and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), no. 2, 309–316. MR 932299, DOI 10.2307/2007055
Andrew John Sommese, Submanifolds of Abelian varieties, Math. Ann. 233 (1978), no. 3, 229–256. MR 466647, DOI 10.1007/BF01405353
Andrew John Sommese and A. Van de Ven, On the adjunction mapping, Math. Ann. 278 (1987), no. 1-4, 593–603. MR 909240, DOI 10.1007/BF01458083
MichałSzurek and Jarosław A. Wiśniewski, Fano bundles of rank $2$ on surfaces, Compositio Math. 76 (1990), no. 1-2, 295–305. Algebraic geometry (Berlin, 1988). MR 1078868
Yun-Gang Ye and Qi Zhang, On ample vector bundles whose adjunction bundles are not numerically effective, Duke Math. J. 60 (1990), no. 3, 671–687. MR 1054530, DOI 10.1215/S0012-7094-90-06027-2