On vector bundles on $3$-folds with sectional genus $1$
HTML articles powered by AMS MathViewer
- by Edoardo Ballico PDF
- Trans. Amer. Math. Soc. 324 (1991), 135-147 Request permission
Abstract:
Here we give a classification (in characteristic zero) of pairs $(V,E)$ with $V$ being a smooth, connected, complete $3$-fold and $E$ a rank-$2$ spanned ample vector bundle on $V$ with sectional genus $1$. The proof uses the partial classification of Fano $3$-folds and Mori theory.References
- Edoardo Ballico, Uniform vector bundles on quadrics, Ann. Univ. Ferrara Sez. VII (N.S.) 27 (1981), 135–146 (1982) (English, with Italian summary). MR 653869
- E. Ballico and P. E. Newstead, Uniform bundles on quadric surfaces and some related varieties, J. London Math. Soc. (2) 31 (1985), no. 2, 211–223. MR 809942, DOI 10.1112/jlms/s2-31.2.211
- W. Barth, Some properties of stable rank-$2$ vector bundles on $\textbf {P}_{n}$, Math. Ann. 226 (1977), no. 2, 125–150. MR 429896, DOI 10.1007/BF01360864
- F. R. DeMeyer and T. J. Ford, On the Brauer group of superspaces and subrings of $k[x,\,y]$, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin-New York, 1982, pp. 211–221. MR 657431
- Takao Fujita, On the structure of polarized manifolds with total deficiency one. I, J. Math. Soc. Japan 32 (1980), no. 4, 709–725. MR 589109, DOI 10.2969/jmsj/03240709 —, Ample vector bundles of small ${c_1}$-sectional genera, preprint.
- Robin Hartshorne, Ample vector bundles on curves, Nagoya Math. J. 43 (1971), 73–89. MR 292847 —, Algebraic geometry, Springer-Verlag, Berlin, 1977.
- Robin Hartshorne, Stable vector bundles of rank $2$ on $\textbf {P}^{3}$, Math. Ann. 238 (1978), no. 3, 229–280. MR 514430, DOI 10.1007/BF01420250
- Paltin Ionescu, Generalized adjunction and applications, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 457–472. MR 830359, DOI 10.1017/S0305004100064409 V. A. Iskoviskih, Fano $3$-folds. I, Math. USSR Izv. 41 (1977), 485-527.
- Sadao Ishimura, On $\pi$-uniform vector bundles, Tokyo J. Math. 2 (1979), no. 2, 337–342. MR 560272, DOI 10.3836/tjm/1270216327
- Yujiro Kawamata, Elementary contractions of algebraic $3$-folds, Ann. of Math. (2) 119 (1984), no. 1, 95–110. MR 736561, DOI 10.2307/2006964
- Shoshichi Kobayashi and Takushiro Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47. MR 316745, DOI 10.1215/kjm/1250523432
- Antonio Lanteri, Marino Palleschi, and Andrew John Sommese, Very ampleness of $K_X\otimes {\scr L}^{\textrm {dim}\,X}$ for ample and spanned line bundles ${\scr L}$, Osaka J. Math. 26 (1989), no. 3, 647–664. MR 1021436
- Robert Lazarsfeld, Some applications of the theory of positive vector bundles, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 29–61. MR 775876, DOI 10.1007/BFb0099356
- Patrick Le Barz, Géométrie énumérative pour les multisécantes, Variétés analytiques compactes (Colloq., Nice, 1977) Lecture Notes in Math., vol. 683, Springer, Berlin, 1978, pp. 116–167 (French). MR 517523 —, Formule multi-secantes pour les courbes gauches quelconques, Enumerative Geometry and Classical Algebraic Geometry, Progress in Math., vol. 24, Birkhäuser, 1982, pp. 165-197.
- Shigefumi Mori and Shigeru Mukai, Classification of Fano $3$-folds with $B_{2}\geq 2$, Manuscripta Math. 36 (1981/82), no. 2, 147–162. MR 641971, DOI 10.1007/BF01170131
- J. P. Murre, Classification of Fano threefolds according to Fano and Iskovskih, Algebraic threefolds (Varenna, 1981) Lecture Notes in Math., vol. 947, Springer, Berlin-New York, 1982, pp. 35–92. MR 672614
- Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
- 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 J. P. Serre, Sur les modules projectif, Séminaire Dubreil-Pisot (1960-61), Secrétariat Math. Paris, Exposé 2, 1961.
- Andrew John Sommese, Submanifolds of Abelian varieties, Math. Ann. 233 (1978), no. 3, 229–256. MR 466647, DOI 10.1007/BF01405353
- A. Van de Ven, On uniform vector bundles, Math. Ann. 195 (1972), 245–248. MR 291182, DOI 10.1007/BF01419597 J. Wisniewski, Length of extremal rays and applications, Ph.D. thesis, University of Notre Dame, 1987.
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 324 (1991), 135-147
- MSC: Primary 14F05; Secondary 14J30
- DOI: https://doi.org/10.1090/S0002-9947-1991-0986021-8
- MathSciNet review: 986021