Quasi-lines and their degenerations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 3, pp. 359-383.

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.

Classification: 14E30, 14J10, 14J30, 14J40, 14J45
@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, Serie 5, Volume 6 (2007) no. 3, pp. 359-383. http://archive.numdam.org/item/ASNSP_2007_5_6_3_359_0/

[1] T. Ando, On extremal rays of the higher-dimensional varieties, Invent. Math. 81 (1985), 347-357. | MR | Zbl

[2] V. Ancona, T. Peternell and J. A. Wiśniewski, Fano bundles and splitting theorems on projective spaces and quadrics, Pacific J. Math. 163 (1994), 17-42. | MR | Zbl

[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] K. Cho, Y. Miyaoka and N. I. Shepherd-Barron, 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

[5] O. Debarre, “Higher-Dimensional Algebraic Geometry”, Universitext, Springer-Verlag, New York, 2001. | MR | Zbl

[6] T. Fujita, 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] R. Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017-1032. | MR | Zbl

[8] R. Hartshorne, “Algebraic Geometry”, Graduate Texts in Mathematics, Vol. 52. Springer-Verlag, New York, 1977. | MR | Zbl

[9] R. Hartshorne, Stable vector bundles of rank 2 on 𝐏 3 , Math. Ann. 238 (1978), 229-280. | MR | Zbl

[10] N. J. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. 43 (1981), 133-150. | MR | Zbl

[11] P. Ionescu and D. Naie, Rationality properties of manifolds containing quasi-lines, Internat. J. Math. 14 (2003), 1053-1080. | MR | Zbl

[12] P. Ionescu and C. Voica, Models of rationally connected manifolds, J. Math. Soc. Japan 55 (2003), 143-164. | MR | Zbl

[13] S. Kebekus, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, In: “Complex geometry” (Göttingen, 2000), Springer, Berlin, 2002, 147-155. | MR | Zbl

[14] Kawamata, Yujiro, Matsuda, Katsumi and Matsuki, Kenji, 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

[15] J. Kollár, “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] S. Mori and S. Mukai, Classification of Fano 3-folds with B 2 2, Manuscripta Math. 36 (1981/82), 147-162. | MR | Zbl

[17] S. Mori and S. Mukai, On Fano 3-folds with B 2 2, In: “Algebraic varieties and analytic varieties” (Tokyo, 1981), Adv. Stud. Pure Math. Vol. 1, North-Holland, Amsterdam, 1983, 101-129. | MR | Zbl

[18] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982), 133-176. | MR | Zbl

[19] C. Okonek, M. Schneider and H. Spindler, “Vector Bundles on Complex Projective Spaces”, Progress in Mathematics, Vol. 3, Birkhäuser Boston, Mass., 1980. | MR | Zbl

[20] W. M. Oxbury, Twistor spaces and Fano threefolds, Quart. J. Math. Oxford Ser. 45 (1994), 343-366. | MR | Zbl

[21] M. Szurek and J. A. Wiśniewski, Fano bundles over 𝐏 3 and Q 3 , Pacific J. Math. 141 (1990), 197-208. | MR | Zbl

[22] M. Szurek and J. A. Wiśniewski, On Fano manifolds, which are 𝐏 k -bundles over 𝐏 2 , Nagoya Math. J. 120 (1990), 89-101. | MR | Zbl

[23] J. A. Wiśniewski, On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math. 417 (1991), 141-157. | MR | Zbl