no. 1
Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence
[Transformations de Crémona quadro-quadriques en basses dimensions via la correspondence JC]
Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 71-111.

Il a été établit précédemment qu’une transformation de Crémona de bidegré (2,2) est linéairement équivalente à la projectivation de l’inversion d’une algèbre de Jordan de rang 3. Ce résultat (appelé la “correspondance JC”) est utilisé dans le présent article pour étudier les transformations birationnelles quadro-quadriques des espaces projectifs de petite dimension. En particulier, nous décrivons de nouvelles familles très simples de telles applications birationnelles et nous obtenons leur classifications complètes et explicites en dimension 4 et 5.

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.

Reçu le :
Accepté le :
DOI : https://doi.org/10.5802/aif.2839
Classification : 14E07,  17Cxx
Mots clés : Transformation de Crémona, Algèbre de Jordan 
@article{AIF_2014__64_1_71_0,
     author = {Pirio, Luc and Russo, Francesco},
     title = {Quadro-quadric {Cremona} transformations in low dimensions via~the~$JC$-correspondence},
     journal = {Annales de l'Institut Fourier},
     pages = {71--111},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {1},
     year = {2014},
     doi = {10.5802/aif.2839},
     zbl = {06387266},
     mrnumber = {3330541},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2839/}
}
Pirio, Luc; Russo, Francesco. Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence. Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 71-111. doi : 10.5802/aif.2839. https://aif.centre-mersenne.org/articles/10.5802/aif.2839/

[1] Allison, B. N. A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann., Tome 237 (1978), pp. 133-156 | Article | MR 507909 | Zbl 0368.17001

[2] Allison, B. N. Simple structurable algebras of skew-dimension one, Comm. Algebra, Tome 18 (1978), pp. 1245-1279 | Article | MR 1059949 | Zbl 0702.17002

[3] Blanc, J. Elements and cyclic subgroups of finite order of the Cremona group, Comment. Math. Helv., Tome 86 (2011), pp. 469-497 | Article | MR 2775137 | Zbl 1213.14029

[4] Braun, H.; Koecher, M. Jordan Algebren, Grund. der Math., Tome 128, Springer-Verlag, 1966 | MR 204470 | Zbl 0145.26001

[5] Bruno, A.; Verra, A. The quadro-quadric Cremona transformations of P 4 and P 5 , Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. (5), Tome 35 (2011), pp. 3-21 | Zbl 1274.14013

[6] Cantat, S.; Lamy, S. Normal subgroups of the Cremona group, Acta Mat., Tome 210 (2013), pp. 31-94 | Article | MR 3037611 | Zbl 1278.14017

[7] Cremona, L. Sulle transformazioni razionali nello spazio, Annali di Mat., Tome 5 (1871-1873), pp. 131-163

[8] Del Pezzo, P. Le trasformazioni coniche dello spazio, Rend. R. Acc. delle Scienze Fisiche e Mat. di Napoli, Tome 2 (1986), pp. 288-296

[9] Del Pezzo, P. Una trasformazione cremoniana fra spazi a quattro dimensioni, Rend. R. Acc. delle Scienze Fisiche e Mat. di Napoli, Tome 2 (1986), pp. 336-344

[10] Ein, L.; Shepherd-Barron, N. Some special Cremona transformations, Amer. J. Math., Tome 111 (1989), pp. 283-800 | Article | MR 1020829 | Zbl 0708.14009

[11] Eisenbud, D.; Harris, J. The geometry of schemes, Grad. Texts in Math., Tome 197, Springer-Verlag, 2000 | MR 1730819 | Zbl 0960.14002

[12] Elgueta, L.; Suazo, A. Jordan nilalgebras of dimension 6, Proyecciones, Tome 21 (2002), pp. 277-289 | Article | MR 1948847

[13] Gabriel, P. Finite representation type is open, Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), Paper No. 10 (1974), 23 p. pages | MR 376769 | Zbl 0313.16034

[14] Gerstenhaber, M.; Myung, H. C. On commutative power-associative nilalgebras of low dimension, Proc. Amer. Math. Soc., Tome 48 (1975), pp. 29-32 | Article | MR 364365 | Zbl 0314.17001

[15] Grayson, D. R.; Stillman, M. E. Macaulay2, a software system for research in algebraic geometry (Available at http://www.math.uiuc.edu/Macaulay2/)

[16] Harris, J. Curves in projective space, Séminaire de Mathématiques Supérieures, Tome 85, Presses Univ. Montréal, 1982 | MR 685427 | Zbl 0511.14014

[17] Hudson, H. P. Cremona transformations in plane and space, Cambridge University Press, 1927

[18] Jacobson, N. Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Tome XXXIX, AMS, 1968 | MR 251099 | Zbl 0218.17010

[19] Kaji, H.; Yasukura, O. Projective geometry of Freudenthal’s varieties of certain type, Michigan Math. J., Tome 52 (2004), pp. 515-542 | Article | MR 2097396 | Zbl 1071.14051

[20] Kashuba, I. Variety of Jordan algebras in small dimensions, Algebra Discrete Math., Tome 2 (2006), pp. 62-76 | MR 2320983 | Zbl 1155.14305

[21] Mallol, C.; Varro, R.; Nourigat, M. Sur la classification des nilalgèbres commutatives de nilindice 3, Algebra Discrete Math., Tome 33 (2005), pp. 4149-4158 | MR 2183988 | Zbl 1094.17001

[22] Marletta Carbonaro, C. I sistemi omaloidici di ipersuperficie dell’ S 4 , legati alle algebre complesse di ordine 4, dotate di modulo, Rend. Accad. Sci. Fis. Mat. Napoli, Tome 15 (1949), pp. 168-201 | MR 65207

[23] Marletta Carbonaro, C. La funzione inversa y=x -1 in una algebra complessa semi–semplice, Boll. Accad. Gioenia Sci. Nat. Catania, Tome 2 (1953), pp. 195-201 | MR 59881

[24] Martin-Deschamps, M.; Piene, R. Arithmetically Cohen-Macaulay curves in P 4 of degree 4 and genus 0, Manuscripta Math., Tome 93 (1997), pp. 391-408 | Article | MR 1457737 | Zbl 0905.14027

[25] McCrimmon, K. A taste of Jordan algebras, Universitext, Springer-Verlag, 2004 | MR 2014924 | Zbl 1044.17001

[26] Notari, R.; Spreafico, M. L. A stratification of Hilbert schemes by initial ideals and applications, Manuscripta Math., Tome 101 (2000), pp. 429-448 | Article | MR 1759253 | Zbl 0985.13006

[27] Pan, I. Sur les transformations de Cremona de bidegré (3,3), Enseign. Math., Tome 43 (1997), pp. 285-297 | MR 1489888 | Zbl 0921.14005

[28] Pan, I.; Ronga, F.; Vust, T. Transformations birationnelles quadratiques de l’espace projectif complexe à trois dimensions, Ann. Inst. Fourier, Tome 51 (2001), pp. 1153-1187 | Article | Numdam | MR 1860661 | Zbl 0987.14009

[29] Pan, I.; Russo, F. Cremona transformations and special double structures, Manuscripta Math., Tome 117 (2005), pp. 491-510 | Article | MR 2163490 | Zbl 1093.14018

[30] Pirio, L. Classification of rank three Jordan algebras of low dimension (in preparation)

[31] Pirio, L.; Russo, F. Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and cubic Jordan algebras (Preprint arXiv:1109.3573)

[32] Pirio, L.; Russo, F. On projective varieties n-covered by curves of degree δ, Comm. Math. Helv., Tome 88 (2013), pp. 715-756 | Article | MR 3093508 | Zbl pre06216015

[33] Schafer, R. D. An introduction to nonassociative algebras, Pure and Applied Mathematics, Tome 22, Academic Press, 1995 | MR 1375235 | Zbl 0145.25601

[34] Scorza, G. Le algebre del 4. ∘ ordine, Atti Accad. Sci. Fis. Mat. Napoli, II. Ser. 20 (1935) no. 14, pp. 1-83 | Zbl 0012.39203

[35] Semple, J. G. Cremona transformations of space of four dimensions by means of quadrics, and the reverse transformations, Philosophical Transactions of the Royal Society of London (1929) no. 228, pp. 331-376 | Article

[36] Semple, J. G.; Roth, L. Introduction to Algebraic Geometry, Oxford University Press, 1949 and 1986 | MR 34048 | Zbl 0576.14001

[37] Snyder, V.; Black, A.; Coble, A.; Dye, L.; Emch, A.; Lefschetz, S.; Sharpe, F. R.; Sisam, C. Selected topics in algebraic geometry, Chelsea Publishing Co., 1970 | MR 290927 | Zbl 0213.47101

[38] Spampinato, N. 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) no. 67, pp. 80-86

[39] Study, E. Über Systeme complexer Zahlen und ihre Anwendung in der Theorie der Transformationsgruppen, Monatsh. f. Math. I. (1890), pp. 283-355 | Article

[40] Wesseler, H. Der Klassifikation der Jordan-Algebren niedriger Dimension (1978) (Staatsexamensarbeit für das Lehramt am Gymnasium)

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