Quadro-quadric Cremona transformations in low dimensions via the  J C -correspondence

Luc Pirio[1]; Francesco Russo[2]

  • [1] IRMAR, UMR 6625 du CNRS, Université Rennes 1, Campus de beaulieu, 35000 Rennes, France
  • [2] Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 1, page 71-111
  • ISSN: 0373-0956

Abstract

top
It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “ J C -correspondence”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.

How to cite

top

Pirio, Luc, and Russo, Francesco. "Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence." Annales de l’institut Fourier 64.1 (2014): 71-111. <http://eudml.org/doc/275574>.

@article{Pirio2014,
abstract = {It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “$JC$-correspondence”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.},
affiliation = {IRMAR, UMR 6625 du CNRS, Université Rennes 1, Campus de beaulieu, 35000 Rennes, France; Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy},
author = {Pirio, Luc, Russo, Francesco},
journal = {Annales de l’institut Fourier},
keywords = {Cremona transformation; Jordan algebra; quadro-quadric Cremona transformations; Jordan algebras; cubo-cubic Cremona transformations},
language = {eng},
number = {1},
pages = {71-111},
publisher = {Association des Annales de l’institut Fourier},
title = {Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence},
url = {http://eudml.org/doc/275574},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Pirio, Luc
AU - Russo, Francesco
TI - Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 1
SP - 71
EP - 111
AB - It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “$JC$-correspondence”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.
LA - eng
KW - Cremona transformation; Jordan algebra; quadro-quadric Cremona transformations; Jordan algebras; cubo-cubic Cremona transformations
UR - http://eudml.org/doc/275574
ER -

References

top
  1. B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 (1978), 133-156 Zbl0368.17001MR507909
  2. B. N. Allison, Simple structurable algebras of skew-dimension one, Comm. Algebra 18 (1978), 1245-1279 Zbl0702.17002MR1059949
  3. J. Blanc, Elements and cyclic subgroups of finite order of the Cremona group, Comment. Math. Helv. 86 (2011), 469-497 Zbl1213.14029MR2775137
  4. H. Braun, M. Koecher, Jordan Algebren, 128 (1966), Springer-Verlag Zbl0145.26001MR204470
  5. A. Bruno, A. Verra, The quadro-quadric Cremona transformations of P 4 and P 5 , Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. (5) 35 (2011), 3-21 Zbl1274.14013
  6. S. Cantat, S. Lamy, Normal subgroups of the Cremona group, Acta Mat. 210 (2013), 31-94 Zbl1278.14017MR3037611
  7. L. Cremona, Sulle transformazioni razionali nello spazio, Annali di Mat. 5 (1871-1873), 131-163 Zbl04.0418.02
  8. P. Del Pezzo, Le trasformazioni coniche dello spazio, Rend. R. Acc. delle Scienze Fisiche e Mat. di Napoli 2 (1986), 288-296 Zbl27.0560.02
  9. P. Del Pezzo, Una trasformazione cremoniana fra spazi a quattro dimensioni, Rend. R. Acc. delle Scienze Fisiche e Mat. di Napoli 2 (1986), 336-344 Zbl27.0540.01
  10. L. Ein, N. Shepherd-Barron, Some special Cremona transformations, Amer. J. Math. 111 (1989), 283-800 Zbl0708.14009MR1020829
  11. D. Eisenbud, J. Harris, The geometry of schemes, 197 (2000), Springer-Verlag Zbl0960.14002MR1730819
  12. L. Elgueta, A. Suazo, Jordan nilalgebras of dimension 6, Proyecciones 21 (2002), 277-289 Zbl1016.17001MR1948847
  13. P. Gabriel, Finite representation type is open, Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), Paper No. 10 (1974), Carleton Univ., Ottawa, Ont. Zbl0313.16034MR376769
  14. M. Gerstenhaber, H. C. Myung, On commutative power-associative nilalgebras of low dimension, Proc. Amer. Math. Soc. 48 (1975), 29-32 Zbl0314.17001MR364365
  15. D. R. Grayson, M. E. Stillman, Macaulay2, a software system for research in algebraic geometry 
  16. J. Harris, Curves in projective space, 85 (1982), Presses Univ. Montréal Zbl0511.14014MR685427
  17. H. P. Hudson, Cremona transformations in plane and space, (1927), Cambridge University Press Zbl53.0595.01
  18. N. Jacobson, Structure and representations of Jordan algebras, XXXIX (1968), AMS Zbl0218.17010MR251099
  19. H. Kaji, O. Yasukura, Projective geometry of Freudenthal’s varieties of certain type, Michigan Math. J. 52 (2004), 515-542 Zbl1071.14051MR2097396
  20. I. Kashuba, Variety of Jordan algebras in small dimensions, Algebra Discrete Math. 2 (2006), 62-76 Zbl1155.14305MR2320983
  21. C. Mallol, R. Varro, M. Nourigat, Sur la classification des nilalgèbres commutatives de nilindice 3, Algebra Discrete Math. 33 (2005), 4149-4158 Zbl1094.17001MR2183988
  22. C. Marletta Carbonaro, I sistemi omaloidici di ipersuperficie dell’ S 4 , legati alle algebre complesse di ordine 4, dotate di modulo, Rend. Accad. Sci. Fis. Mat. Napoli 15 (1949), 168-201 MR65207
  23. C. Marletta Carbonaro, La funzione inversa y = x - 1 in una algebra complessa semi–semplice, Boll. Accad. Gioenia Sci. Nat. Catania 2 (1953), 195-201 MR59881
  24. M. Martin-Deschamps, R. Piene, Arithmetically Cohen-Macaulay curves in P 4 of degree 4 and genus 0 , Manuscripta Math. 93 (1997), 391-408 Zbl0905.14027MR1457737
  25. K. McCrimmon, A taste of Jordan algebras, (2004), Springer-Verlag Zbl1044.17001MR2014924
  26. R. Notari, M. L. Spreafico, A stratification of Hilbert schemes by initial ideals and applications, Manuscripta Math. 101 (2000), 429-448 Zbl0985.13006MR1759253
  27. I. Pan, Sur les transformations de Cremona de bidegré ( 3 , 3 ) , Enseign. Math. 43 (1997), 285-297 Zbl0921.14005MR1489888
  28. I. Pan, F. Ronga, T. Vust, Transformations birationnelles quadratiques de l’espace projectif complexe à trois dimensions, Ann. Inst. Fourier 51 (2001), 1153-1187 Zbl0987.14009MR1860661
  29. I. Pan, F. Russo, Cremona transformations and special double structures, Manuscripta Math. 117 (2005), 491-510 Zbl1093.14018MR2163490
  30. L. Pirio, Classification of rank three Jordan algebras of low dimension Zbl1065.53018
  31. L. Pirio, F. Russo, Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and cubic Jordan algebras Zbl1311.14017
  32. L. Pirio, F. Russo, On projective varieties n -covered by curves of degree δ , Comm. Math. Helv. 88 (2013), 715-756 Zbl1298.14055MR3093508
  33. R. D. Schafer, An introduction to nonassociative algebras, 22 (1995), Academic Press Zbl0145.25601MR1375235
  34. G. Scorza, Le algebre del 4 . ∘ ordine, Atti Accad. Sci. Fis. Mat. Napoli, II. Ser. 20 (1935), 1-83 Zbl0012.39203
  35. J. G. Semple, Cremona transformations of space of four dimensions by means of quadrics, and the reverse transformations, Philosophical Transactions of the Royal Society of London (1929), 331-376 Zbl55.0991.02
  36. J. G. Semple, L. Roth, Introduction to Algebraic Geometry, (1949 and 1986), Oxford University Press Zbl0576.14001MR34048
  37. V. Snyder, A. Black, A. Coble, L. Dye, A. Emch, S. Lefschetz, F. R. Sharpe, C. Sisam, Selected topics in algebraic geometry, (1970), Chelsea Publishing Co. Zbl0213.47101MR290927
  38. N. Spampinato, I gruppi di affinità e di trasformazioni quadratiche piane legati alle due algebre complesse doppie dotate di modulo, Boll. Accad. Gioenia Sci. Nat. Catania (1935), 80-86 Zbl61.1034.02
  39. E. Study, Über Systeme complexer Zahlen und ihre Anwendung in der Theorie der Transformationsgruppen, Monatsh. f. Math. I. (1890), 283-355 
  40. H. Wesseler, Der Klassifikation der Jordan-Algebren niedriger Dimension, (1978) 

NotesEmbed ?

top

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.