Abstract
We give a complete list of square-free monomial Cremona maps of \({\mathbb {P}}^{n-1}\), with \(n\le 6\), up to equivalence classes. Also, we present a sufficient condition for all the ideals associated to a square-free Cremona transformation with same degree d in a fixed projective space \({\mathbb {P}}^{n-1}\) to have eight two. Finally, we introduce an algorithm to count them. Using this algorithm, we obtain the complete list of square-free Cremona maps of \({\mathbb {P}}^{n-1}\) when the degree is two for \(n=7,n=8,n=9, n=10, n=11\) and when the degree is three for \(n=7\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alberich-Carramiñana, M.: Geometry of the Plane Cremona Maps. Lecture Notes in Mathematics, vol. 1769. Springer-Verlag, Berlin (2002)
Costa, B., Simis, A.: Cremona maps defined by monomials. J. Pure Appl. Algebra 216, 202–215 (2012)
Costa, B., Simis, A.: New constructions of cremona maps. Math. Res. Lett. 20, 629–645 (2013)
Dória, A., Simis, A.: The Newton complementary dual revisited. J. Algebra Appl. 17, 1850004–1850016 (2017)
Gonzalez-Sprinberg, G., Pan, I.: On the monomial birational maps of the projective space. An. Acad. Brasil. Ciênc. 75, 129–134 (2003)
Korchagin, A.B.: On birational monomial transformations of plane. Int. J. Math. Math. Sci. 32, 1671–1677 (2004)
Pirio, L., Russo, F.: Quadro-quadric Cremona transformations in low dimensions via the J C-correspondence. Ann. de l’inst. Fourier. 64(1), 71–111 (2014)
Pirio, L., Russo, F.: The XJC-correspondence. Journal für die reine und angewandte Mathematik (Crelles Journal). 2016(716), 229–250 (2016)
Ramos, Z., Simis, A.: De Jonquières transformations in arbitrary dimension. An ideal theoretic view. Commut. Algebra, pp 1–14 (2022)
Simis, A.: Cremona transformations and some related algebras. J. Algebra 280, 162–179 (2004)
Simis, A., Villarreal, R.H.: Constraints for the normality of monomial subrings and birationality. Proc. Am. Math. Soc. 131, 2043–2048 (2003)
Simis, A., Villarreal, R.H.: Linear syzygies and birational combinatorics. Results Math. 48(3–4), 326–343 (2005)
Simis, A., Villarreal, R.H.: Combinatorics of Cremona monomial maps. Math. Comput. 81(279), 1857–1867 (2012)
Villarreal, R.H.: Monomial Algebras, Monographs and Textbooks in Pure and Applied Mathematics 238. Marcel Dekker, New York (2001)
Acknowledgements
The authors dedicate this paper to Rafael H. Villarreal on the occasion of his 70th birthday. We thank Aron Simis for his valuable support during this research and the referee for carefully reading our manuscript and their constructive comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Communicated by Isidoro Gitler.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Costa, B., Dias, T., Gondim, R. et al. Counting square-free monomial Cremona maps. São Paulo J. Math. Sci. 17, 76–101 (2023). https://doi.org/10.1007/s40863-022-00345-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40863-022-00345-5