A classification theorem on Fano bundles

Muñoz, Roberto; Solá Conde, Luis E.; Occhetta, Gianluca

Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 341-373.

Dans cet article, on classifie les fibrés de Fano de rang deux $ℰ$ sur les variétés de Fano satisfaisant $H^2(X,\mathbb{Z})\cong H^4(X,\mathbb{Z})\cong \mathbb{Z}$. La classification est obtenue par le calcul des cônes nef et pseudoeffectif de la projectivation $\mathbb{P}\left(ℰ\right)$, ce qui nous permet d’obtenir des invariants cohomologiques de $X$ et $ℰ$. Comme un sous-produit, nous discutons des fibrés associés à Fano congruences de droites, montrant que leurs variétés de tangentes rationnelles minimales peuvent avoir plusieurs composants linéaires.

In this paper we classify rank two Fano bundles $ℰ$ on Fano manifolds satisfying $H^2(X,\mathbb{Z})\cong H^4(X,\mathbb{Z})\cong \mathbb{Z}$. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $\mathbb{P}\left(ℰ\right)$, that allows us to obtain the cohomological invariants of $X$ and $ℰ$. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

[1] Ancona, V.; Peternell, T.; Wiśniewski, J. A. Fano bundles and splitting theorems on projective spaces and quadrics, Pacific J. Math., Volume 163 (1994) no. 1, pp. 17-42 http://projecteuclid.org/getRecord?id=euclid.pjm/1102622627 | DOI | MR | Zbl

[2] Andreatta, M.; Ballico, E.; Wiśniewski, J. A. Two theorems on elementary contractions, Math. Ann., Volume 297 (1993) no. 2, pp. 191-198 | DOI | MR | Zbl

[3] Bazan, D.; Mezzetti, E. On the construction of some Buchsbaum varieties and the Hilbert scheme of elliptic scrolls in ${\mathbb{P}}^5$, Geom. Dedicata, Volume 86 (2001) no. 1-3, pp. 191-204 | DOI | MR | Zbl

[4] Campana, F.; Peternell, T. Projective manifolds whose tangent bundles are numerically effective, Math. Ann., Volume 289 (1991) no. 1, pp. 169-187 | DOI | MR | Zbl

[5] Cheltsov, I. A. Conic bundles with big discriminant loci, Izv. Ross. Akad. Nauk Ser. Mat., Volume 68 (2004) no. 2, pp. 215-221 | DOI | MR | Zbl

[6] Cornalba, M. A remark on the topology of cyclic coverings of algebraic varieties, Boll. Un. Mat. Ital. A (5), Volume 18 (1981) no. 2, pp. 323-328 | MR | Zbl

[7] De Poi, P. Threefolds in ${\mathbb{P}}^5$ with one apparent quadruple point, Comm. Algebra, Volume 31 (2003) no. 4, pp. 1927-1947 | DOI | MR | Zbl

[8] Ein, L.; Shepherd-Barron, N. Some special Cremona transformations, Amer. J. Math., Volume 111 (1989) no. 5, pp. 783-800 | DOI | MR | Zbl

[9] Fujita, T. Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, 155, Cambridge University Press, Cambridge, 1990 | DOI | MR | Zbl

[10] Griffiths, P.; Harris, J. Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978, pp. xii+813 | MR | Zbl

[11] Hulek, K. Stable rank-$2$ vector bundles on ${\mathbb{P}}_2$ with $c_1$ odd, Math. Ann., Volume 242 (1979) no. 3, pp. 241-266 | DOI | EuDML | MR | Zbl

[12] Hwang, J.-M. Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) (ICTP Lect. Notes), Volume 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 335-393 | MR | Zbl

[13] Hwang, J.-M. On the degrees of Fano four-folds of Picard number 1, J. Reine Angew. Math., Volume 556 (2003), pp. 225-235 | DOI | MR | Zbl

[14] Hwang, J.-M. Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1, Ann. Inst. Fourier (Grenoble), Volume 57 (2007) no. 3, pp. 815-823 | DOI | EuDML | Numdam | MR | Zbl

[15] Hwang, J.-M.; Mok, N. Birationality of the tangent map for minimal rational curves, Asian J. Math., Volume 8 (2004) no. 1, pp. 51-63 | DOI | MR | Zbl

[16] Iliev, A.; Manivel, L. Severi varieties and their varieties of reductions, J. Reine Angew. Math., Volume 585 (2005), pp. 93-139 | DOI | MR | Zbl

[17] Lazarsfeld, R. A Barth-type theorem for branched coverings of projective space, Math. Ann., Volume 249 (1980) no. 2, pp. 153-162 | DOI | EuDML | MR | Zbl

[18] Maruyama, M. Boundedness of semistable sheaves of small ranks, Nagoya Math. J., Volume 78 (1980), pp. 65-94 http://projecteuclid.org/getRecord?id=euclid.nmj/1118786090 | MR | Zbl

[19] Mukai, S. Biregular classification of Fano $3$-folds and Fano manifolds of coindex $3$, Proc. Nat. Acad. Sci. U.S.A., Volume 86 (1989) no. 9, pp. 3000-3002 | DOI | MR | Zbl

[20] Muñoz, Roberto; Occhetta, Gianluca; Solá Conde, Luis E. On rank $2$ vector bundles on Fano manifolds, 2011 (preprint math.AG/1104.1490. To appear in Kyoto J. Math.) | Zbl

[21] Muñoz, Roberto; Occhetta, Gianluca; Solá Conde, Luis E. Rank two Fano bundles on $𝔾(1,4)$, J. Pure Appl. Algebra, Volume 216 (2012) no. 10, pp. 2269-2273 | DOI | MR | Zbl

[22] Muñoz, Roberto; Occhetta, Gianluca; Solá Conde, Luis E. Uniform vector bundles on Fano manifolds and applications, J. Reine Angew. Math., Volume 664 (2012), pp. 141-162 | MR | Zbl

[23] Niven, I. Irrational numbers, The Carus Mathematical Monographs, No. 11, The Mathematical Association of America. John Wiley and Sons, Inc., New York, N.Y., 1956 | MR | Zbl

[24] Novelli, C.; Occhetta, Gianluca Projective manifolds containing a large linear subspace with nef normal bundle, Michigan Math. J., Volume 60 (2011) no. 2, pp. 441-462 | DOI | MR | Zbl

[25] Okonek, C.; Schneider, M.; Spindler, H. Vector bundles on complex projective spaces, Progress in Mathematics, 3, Birkhäuser, Boston, Mass., 1980, pp. viii+239 | MR | Zbl

[26] Ottaviani, G. On Cayley bundles on the five-dimensional quadric, Boll. Un. Mat. Ital. A (7), Volume 4 (1990) no. 1, pp. 87-100 | MR | Zbl

[27] Reid, M. The complete intersection of two or more quadrics, University of Cambridge (1972) (Ph. D. Thesis)

[28] Sarkisov, V. G. On conic bundle structures, Izv. Akad. Nauk SSSR Ser. Mat., Volume 46 (1982) no. 2, p. 371-408, 432 | MR | Zbl

[29] Sato, E. Projective manifolds swept out by large-dimensional linear spaces, Tohoku Math. J. (2), Volume 49 (1997) no. 3, pp. 299-321 | DOI | MR | Zbl

[30] Sols, Ignacio; Szurek, Michał; Wiśniewski, Jarosław A. Rank-$2$ Fano bundles over a smooth quadric $Q_3$, Pacific J. Math., Volume 148 (1991) no. 1, pp. 153-159 http://projecteuclid.org/getRecord?id=euclid.pjm/1102644787 | DOI | MR | Zbl

[31] Szurek, Michał; Wiśniewski, Jarosław A. Fano bundles over ${\mathbb{P}}^3$ and ${\mathbb{Q}}_3$, Pacific J. Math., Volume 141 (1990) no. 1, pp. 197-208 http://projecteuclid.org/getRecord?id=euclid.pjm/1102646779 | DOI | Zbl

[32] Szurek, Michał; Wiśniewski, Jarosław A. On Fano manifolds, which are ${\mathbb{P}}^k$-bundles over ${\mathbb{P}}^2$, Nagoya Math. J., Volume 120 (1990), pp. 89-101 http://projecteuclid.org/getRecord?id=euclid.nmj/1118782199 | MR | Zbl

[33] Wiśniewski, Jarosław A. On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math., Volume 417 (1991), pp. 141-157 | DOI | EuDML | MR | Zbl

[34] Zak, F. L. Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, 127, American Mathematical Society, Providence, RI, 1993 | MR | Zbl