Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence
Annales de l'Institut Fourier, Volume 64 (2014) no. 1, pp. 71-111.

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.

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.

DOI: 10.5802/aif.2839
Classification: 14E07, 17Cxx
Keywords: Cremona transformation, Jordan algebra
Mot clés : Transformation de Crémona, Algèbre de Jordan
Pirio, Luc 1; Russo, Francesco 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
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Pirio, Luc; Russo, Francesco. Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence. Annales de l'Institut Fourier, Volume 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., Volume 237 (1978), pp. 133-156 | DOI | MR | Zbl

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

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

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

[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), Volume 35 (2011), pp. 3-21 | Zbl

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

[7] Cremona, L. Sulle transformazioni razionali nello spazio, Annali di Mat., Volume 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, Volume 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, Volume 2 (1986), pp. 336-344

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

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

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

[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), pp. 23 p. | MR | Zbl

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

[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, 85, Presses Univ. Montréal, 1982 | MR | Zbl

[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, XXXIX, AMS, 1968 | MR | Zbl

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

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

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

[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, Volume 15 (1949), pp. 168-201 | MR

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

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

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

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

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

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

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

[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., Volume 88 (2013), pp. 715-756 | DOI | MR

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

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

[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 | DOI

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

[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 | Zbl

[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 | DOI

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

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