Combinatorics of Cremona monomial maps
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- by Aron Simis and Rafael H. Villarreal PDF
- Math. Comp. 81 (2012), 1857-1867 Request permission
Abstract:
We study Cremona monomial maps using linear algebra, lattice theory and linear optimization methods. Among the results is a simple integer matrix theoretic proof that the inverse of a Cremona monomial map is also defined by monomials of fixed degree, and moreover, the set of monomials defining the inverse can be obtained explicitly in terms of the initial data. We present another method to compute the inverse of a Cremona monomial map based on integer programming techniques and the notion of a Hilbert basis. A neat consequence is drawn for the plane Cremona monomial group, in particular, the known result saying that a plane Cremona monomial map and its inverse have the same degree.References
- Maria Alberich-Carramiñana, Geometry of the plane Cremona maps, Lecture Notes in Mathematics, vol. 1769, Springer-Verlag, Berlin, 2002. MR 1874328, DOI 10.1007/b82933
- W. G. Bridges and H. J. Ryser, Combinatorial designs and related systems, J. Algebra 13 (1969), 432–446. MR 245456, DOI 10.1016/0021-8693(69)90085-4
- W. Bruns and B. Ichim, Normaliz 2.0, Computing normalizations of affine semigroups 2008. Available from http://www.math.uos.de/normaliz.
- Gérard Cornuéjols, Combinatorial optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Packing and covering. MR 1828452, DOI 10.1137/1.9780898717105
- B. Costa and A. Simis, Cremona maps defined by monomials. J. Pure Appl. Algebra, In Press, DOI:10.1016/j.jpaa.2011.06.007, 2011, arXiv:1101.2413.
- Gérard Gonzalez-Sprinberg and Ivan Pan, On the monomial birational maps of the projective space, An. Acad. Brasil. Ciênc. 75 (2003), no. 2, 129–134 (English, with English and Portuguese summaries). MR 1984551, DOI 10.1590/S0001-37652003000200001
- Anatoly B. Korchagin, On birational monomial transformations of plane, Int. J. Math. Math. Sci. 29-32 (2004), 1671–1677. MR 2085087, DOI 10.1155/S0161171204306514
- Alexander Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication. MR 874114
- Aron Simis, Cremona transformations and some related algebras, J. Algebra 280 (2004), no. 1, 162–179. MR 2081926, DOI 10.1016/j.jalgebra.2004.03.025
- Aron Simis and Rafael H. Villarreal, Constraints for the normality of monomial subrings and birationality, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2043–2048. MR 1963748, DOI 10.1090/S0002-9939-02-06790-4
- Aron Simis and Rafael H. Villarreal, Linear syzygies and birational combinatorics, Results Math. 48 (2005), no. 3-4, 326–343. MR 2215584, DOI 10.1007/BF03323372
Additional Information
- Aron Simis
- Affiliation: Departamento de Matemática, Universidade Federal de Pernambuco, 50740-540 Recife, Pe, Brazil
- MR Author ID: 162400
- Email: aron@dmat.ufpe.br
- Rafael H. Villarreal
- Affiliation: Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14–740, 07000 Mexico City, D.F.
- Email: vila@math.cinvestav.mx
- Received by editor(s): September 1, 2009
- Received by editor(s) in revised form: April 5, 2011
- Published electronically: October 24, 2011
- Additional Notes: The first author was partially supported by a grant of CNPq. He warmly thanks CINVESTAV for support during a visit. The second author was partially supported by CONACyT grant 49251-F and SNI
- © Copyright 2011 American Mathematical Society
- Journal: Math. Comp. 81 (2012), 1857-1867
- MSC (2010): Primary 14E05, 14E07, 15A51, 15A36
- DOI: https://doi.org/10.1090/S0025-5718-2011-02556-1
- MathSciNet review: 2904605