Nous construisons des familles d’involutions birationnelles sur ou sur une cubique lisse de dimension qui ne s’intègrent pas dans une relation élémentaire non triviale de liens de Sarkisov. En conséquence, nous construisons de nouveaux homomorphismes à partir de leur groupe de transformations birationnelles, redémontrant de manière effective leur non-simplicité. Nous prouvons également que ces groupes admettent une structure de produit libre. Enfin, nous produisons des automorphismes de ces groupes qui ne sont pas engendrés par des automorphismes intérieurs et des automorphismes de corps.
We construct families of birational involutions on or on a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.
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@article{JEP_2023__10__233_0,
author = {Sokratis Zikas},
title = {Rigid birational involutions of $\mathbb{P}^3$ and cubic~threefolds},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {233--252},
publisher = {\'Ecole polytechnique},
volume = {10},
year = {2023},
doi = {10.5802/jep.217},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.217/}
}
TY - JOUR
AU - Sokratis Zikas
TI - Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds
JO - Journal de l’École polytechnique — Mathématiques
PY - 2023
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EP - 252
VL - 10
PB - École polytechnique
UR - https://jep.centre-mersenne.org/articles/10.5802/jep.217/
DO - 10.5802/jep.217
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Sokratis Zikas. Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 233-252. doi : 10.5802/jep.217. https://jep.centre-mersenne.org/articles/10.5802/jep.217/
[BL12] - “Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links”, Proc. London Math. Soc. (3) 105 (2012) no. 5, p. 1047-1075 | DOI | Zbl
[BL15] - “On birational maps from cubic threefolds”, North-West. Eur. J. Math. 1 (2015), p. 55-84
[BLZ21] - “Quotients of higher-dimensional Cremona groups”, Acta Math. 226 (2021) no. 2, p. 211-318 | DOI | Zbl
[BY20] - “Quotients of groups of birational transformations of cubic del Pezzo fibrations”, J. Éc. polytech. Math. 7 (2020), p. 1089-1112 | DOI | Numdam | Zbl
[CL13] - “Normal subgroups in the Cremona group”, Acta Math. 210 (2013) no. 1, p. 31-94, With an appendix by Yves de Cornulier | DOI | Zbl
[CM13] - “Towards the classification of weak Fano threefolds with ”, Cent. Eur. J. Math. 11 (2013) no. 9, p. 1552-1576 | DOI | Zbl
[Cor95] - “Factoring birational maps of threefolds after Sarkisov”, J. Algebraic Geom. 4 (1995) no. 2, p. 223-254 | Zbl
[Deb01] - Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001 | DOI
[Dés06] - “Sur les automorphismes du groupe de Cremona”, Compositio Math. 142 (2006) no. 6, p. 1459-1478 | DOI | Zbl
[Fuj99] - “Applications of Kawamata’s positivity theorem”, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999) no. 6, p. 75-79 | Zbl
[Har77] - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York-Heidelberg, 1977 | DOI
[HM13] - “The Sarkisov program”, J. Algebraic Geom. 22 (2013) no. 2, p. 389-405 | DOI | Zbl
[Hur92] - “Ueber algebraische Gebilde mit eindeutigen Transformationen in sich”, Math. Ann. 41 (1892) no. 3, p. 403-442 | DOI | Zbl
[IP99] - “Fano varieties”, in Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, p. 1-247 | Zbl
[Isk96] - “Factorization of birational mappings of rational surfaces from the point of view of Mori theory”, Uspekhi Mat. Nauk 51 (1996) no. 4(310), p. 3-72 | DOI
[Kal13] - “Relations in the Sarkisov program”, Compositio Math. 149 (2013) no. 10, p. 1685-1709 | DOI | Zbl
[Kaw01] - “Divisorial contractions in dimension three which contract divisors to smooth points”, Invent. Math. 145 (2001) no. 1, p. 105-119 | DOI | Zbl
[Lon16] - “Non simplicité du groupe de Cremona sur tout corps”, Ann. Inst. Fourier (Grenoble) 66 (2016) no. 5, p. 2021-2046 | DOI | Numdam | Zbl
[LZ20] - “Signature morphisms from the Cremona group over a non-closed field”, J. Eur. Math. Soc. (JEMS) 22 (2020) no. 10, p. 3133-3173 | DOI | Zbl
[MM64] - “On the automorphisms of hypersurfaces”, J. Math. Kyoto Univ. 3 (1963/64), p. 347-361 | DOI
[Sch22] - “Relations in the Cremona group over perfect fields”, Ann. Inst. Fourier (Grenoble) 72 (2022) no. 1, p. 1-42 | DOI | Zbl
[Tzi03] - “Terminal 3-fold divisorial contractions of a surface to a curve. I”, Compositio Math. 139 (2003) no. 3, p. 239-261 | DOI | Zbl
[UZ21] - “Continuous automorphisms of Cremona groups”, Internat. J. Math. 32 (2021) no. 4, article ID 2150019, 17 pages | DOI | Zbl
[Zik23] - “Sarkisov links with centres space curves on smooth cubic surfaces”, Publ. Mat. 67 (2023) no. 2
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