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 \(n=7\).
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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.
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Communicated by Isidoro Gitler.
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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
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DOI: https://doi.org/10.1007/s40863-022-00345-5