General elephants of -Fano 3-folds

Valery Alexeev

Compositio Mathematica (1994)

  • Volume: 91, Issue: 1, page 91-116
  • ISSN: 0010-437X

How to cite


Alexeev, Valery. "General elephants of $\mathbb {Q}$-Fano 3-folds." Compositio Mathematica 91.1 (1994): 91-116. <>.

author = {Alexeev, Valery},
journal = {Compositio Mathematica},
keywords = {-Fano 3-folds; general elephant; Du Val singularities; anti- canonical system; Mori's minimal model program},
language = {eng},
number = {1},
pages = {91-116},
publisher = {Kluwer Academic Publishers},
title = {General elephants of $\mathbb \{Q\}$-Fano 3-folds},
url = {},
volume = {91},
year = {1994},

AU - Alexeev, Valery
TI - General elephants of $\mathbb {Q}$-Fano 3-folds
JO - Compositio Mathematica
PY - 1994
PB - Kluwer Academic Publishers
VL - 91
IS - 1
SP - 91
EP - 116
LA - eng
KW - -Fano 3-folds; general elephant; Du Val singularities; anti- canonical system; Mori's minimal model program
UR -
ER -


  1. [A] V. Alexeev, Ample Weil divisors on K3 surfaces with Du Val singularities, Duke Math. J.64 (1991), 617-624. Zbl0766.14030MR1141287
  2. [Ar] M. Artin, On isolated rational singularities of surfaces, Am. J. Math.88 (1966), 129-137. Zbl0142.18602MR199191
  3. [CM] A. Conte - J.P. Murre, Algebraic varieties of dimension three whose hyperplane sections are Enriques surfaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser. 12 (1985), 43-80. Zbl0612.14041MR818801
  4. [CT] J.-L. Colliot-Thélène, Arithmétique des variétés rationelles et problèmes birationelles, In: "Proc. Int. Conf. Math.", 1986, 641-653. Zbl0698.14060MR934267
  5. [Co] A. Corti, Thesis, in preparation. 
  6. [Fa] G. Fano, Sulle varietà algebriche a tre dimensioni le cui sezioni iperpiane sono superficie di genere zero e bigenere uno, Memoire società dei XL, 24 (1938), 41-66. Zbl0022.07702JFM65.1403.02
  7. [Fl1] A.R. Fletcher, Letter to V. A. Iskovskih, 1987. 
  8. [Fl2] A.R. Fletcher, Working with weighted complete intersections, Max-Plank Institute preprint, 1989. 
  9. [I1] V.A. Iskovskih, Fano threefolds I, II, Izv. Akad. Nauk SSSR, 41 (1977), 516-562, resp. 42 (1978), 506-549. Zbl0407.14016MR463151
  10. [I2] V.A. Iskovskih, Lectures on Algebraic Threefolds. Fano Varieties, Publ. by Moscow University, 1988. 
  11. [Kaw1] Y. Kawamata, Boundedness of Q-Fano threefolds, preprint, 1989. MR1175897
  12. [Kaw2] Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math.127 (1988), 93-163. Zbl0651.14005MR924674
  13. [Kaw3] Y. Kawamata, The minimal discrepancy coefficients of terminal singularities in dimension 3. Appendix to [Sh3]. 
  14. [Kaw4] Y. Kawamata, Termination of log flips for algebraic 3-folds, preprint, 1991. Zbl0814.14016MR1189678
  15. [KMM] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the Minimal Model Program, In: "Algebraic Geometry, Sendai", Adv. Stud. Pure Math.10 (1987), 283-360. Zbl0672.14006MR946243
  16. [Kd] K. Kodaira, On Compact Analytic Surfaces, I. II, III, In "Collected works ", v. 3, Iwanami Shoten, Publishers and Princeton University Press, 1975. Zbl0315.01027MR140519
  17. [Ko1] J. Kollár, Flops, Nagoya Math. J., 113 (1989), 14-36. Zbl0645.14004MR986434
  18. [Ko2] J. Kollár, Flips, flops, minimal models etc, preprint, 1990. MR1144527
  19. [Ma] Yu. I. Manin, Cubic forms. 
  20. [M1] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math.116 (1982), 133-176. Zbl0557.14021MR662120
  21. [M2] S. Mori, Classification of Higher Dimensional Varieties, In: "Algebraic geometry Bowdoin 1985", Proc. Symp. Pure Math.46 (1987), 269-332. Zbl0656.14022MR927961
  22. [M3] S. Mori, Lectures on Fano 3-folds, University of Utah, 1988. 
  23. [N] V.V. Nikulin, Linear systems on singular K3 surfaces, preprint, 1990. 
  24. [R1] M. Reid, Projective morphisms according to Kawamata, preprint, 1983. MR717617
  25. [R2] M. Reid, Young Person's Guide to Canonical Singularities, In: "Algebraic geometry Bowdoin 1985", Proc. Symp. Pure Math.46AMS, 1987, 345-416. Zbl0634.14003MR927963
  26. [R3] M. Reid, Birational geometry of 3-folds according to Sarkisov, unpublished lecture notes, 1991. 
  27. [Sh1] V.V. Shokurov, Smoothness of a general anticanonical divisor on a Fano variety, Izv. Akad. Nauk SSSR, 14 (1980), 395-405. Zbl0429.14012
  28. [Sh2] V.V. Shokurov, The nonvanishing theorem, Izv. Akad. Nauk SSSR, 49 (1985), 635-651. Zbl0605.14006MR794958
  29. [Sh3] V.V. Shokurov, 3-fold log flips, preprint, 1991 (translated by M. Reid). 
  30. [U] T. Urabe, Fixed components of linear systems on K3 surfaces, preprint, 1990. 
  31. [W1] P.M.H. Wilson, Varieties, Bull. London Math. Soc., 19 (1987), 1-48. Zbl0612.14033MR865038
  32. [W2] P.M.H. Wilson, Toward a birational classification of algebraic varieties, Bull. London Math. Soc., 19 (1987), 1-48. Zbl0612.14033MR865038
  33. [KoMo] J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc., 5 (1992), 533-703. Zbl0773.14004MR1149195

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.