In this paper we study the structure of manifolds that contain a quasi-line and give some evidence towards the fact that the irreducible components of degenerations of the quasi-line should determine the Mori cone. We show that the minimality with respect to a quasi-line yields strong restrictions on fibre space structures of the manifold.
@article{ASNSP_2007_5_6_3_359_0, author = {Bonavero, Laurent and H\"oring, Andreas}, title = {Quasi-lines and their degenerations}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {359--383}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {3}, year = {2007}, mrnumber = {2370265}, zbl = {1139.14017}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2007_5_6_3_359_0/} }
TY - JOUR AU - Bonavero, Laurent AU - Höring, Andreas TI - Quasi-lines and their degenerations JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2007 SP - 359 EP - 383 VL - 6 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2007_5_6_3_359_0/ LA - en ID - ASNSP_2007_5_6_3_359_0 ER -
%0 Journal Article %A Bonavero, Laurent %A Höring, Andreas %T Quasi-lines and their degenerations %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 359-383 %V 6 %N 3 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2007_5_6_3_359_0/ %G en %F ASNSP_2007_5_6_3_359_0
Bonavero, Laurent; Höring, Andreas. Quasi-lines and their degenerations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 3, pp. 359-383. http://archive.numdam.org/item/ASNSP_2007_5_6_3_359_0/
[1] On extremal rays of the higher-dimensional varieties, Invent. Math. 81 (1985), 347-357. | MR | Zbl
,[2] Fano bundles and splitting theorems on projective spaces and quadrics, Pacific J. Math. 163 (1994), 17-42. | MR | Zbl
, and ,[3] L. B ădescu, M. C. Beltrametti and P. Ionescu, Almost-lines and quasi-lines on projective manifolds, In: “Complex analysis and algebraic geometry”, de Gruyter, Berlin, 2000, 1-27. | MR | Zbl
[4] Characterizations of projective space and applications to complex symplectic manifolds, In: “Higher dimensional birational geometry” (Kyoto, 1997), Adv. Stud. Pure Math., Vol. 35, Math. Soc. Japan, Tokyo, 2002, 1-88. | MR | Zbl
, and ,[5] “Higher-Dimensional Algebraic Geometry”, Universitext, Springer-Verlag, New York, 2001. | MR | Zbl
,[6] On polarized manifolds whose adjoint bundles are not semipositive, In: “Algebraic geometry” (Sendai, 1985), Adv. Stud. Pure Math., Vol. 10, North-Holland, Amsterdam, 1987, 167-178. | MR | Zbl
,[7] Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017-1032. | MR | Zbl
,[8] “Algebraic Geometry”, Graduate Texts in Mathematics, Vol. 52. Springer-Verlag, New York, 1977. | MR | Zbl
,[9] Stable vector bundles of rank on , Math. Ann. 238 (1978), 229-280. | MR | Zbl
,[10] Kählerian twistor spaces, Proc. London Math. Soc. 43 (1981), 133-150. | MR | Zbl
,[11] Rationality properties of manifolds containing quasi-lines, Internat. J. Math. 14 (2003), 1053-1080. | MR | Zbl
and ,[12] Models of rationally connected manifolds, J. Math. Soc. Japan 55 (2003), 143-164. | MR | Zbl
and ,[13] Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, In: “Complex geometry” (Göttingen, 2000), Springer, Berlin, 2002, 147-155. | MR | Zbl
,[14] Introduction to the minimal model problem, In: “Algebraic Geometry” (Sendai, 1985), Adv. Stud. Pure Math., Vol. 10, North-Holland, Amsterdam, 1987, 283-360. | MR | Zbl
, and ,[15] “Rational Curves on Algebraic Varieties”, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 32, Springer-Verlag, Berlin, 1996. | MR | Zbl
,[16] Classification of Fano -folds with , Manuscripta Math. 36 (1981/82), 147-162. | MR | Zbl
and ,[17] On Fano -folds with , In: “Algebraic varieties and analytic varieties” (Tokyo, 1981), Adv. Stud. Pure Math. Vol. 1, North-Holland, Amsterdam, 1983, 101-129. | MR | Zbl
and ,[18] Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982), 133-176. | MR | Zbl
,[19] “Vector Bundles on Complex Projective Spaces”, Progress in Mathematics, Vol. 3, Birkhäuser Boston, Mass., 1980. | MR | Zbl
, and ,[20] Twistor spaces and Fano threefolds, Quart. J. Math. Oxford Ser. 45 (1994), 343-366. | MR | Zbl
,[21] Fano bundles over and , Pacific J. Math. 141 (1990), 197-208. | MR | Zbl
and ,[22] On Fano manifolds, which are -bundles over , Nagoya Math. J. 120 (1990), 89-101. | MR | Zbl
and ,[23] On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math. 417 (1991), 141-157. | MR | Zbl
,