Cremona maps defined by monomials

Mathematical Research Letters

2013

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Abstract. One defines two ways of constructing rational maps derived from other rational maps, in a characteristic-free context. The first introduces the Newton complementary dual of a rational map. One main result is that this dual preserves birationality and gives an involutional map of the Cremona group to itself that restricts to the monomial Cremona subgroup and preserves de Jonquières maps. In the monomial restriction, this duality commutes with taking inverse in the group, but is a not a group homomorphism. The second construction is an iterative process to obtain rational maps in increasing dimension. Starting with birational maps, it leads to rational maps whose topological degree is under control. Making use of monoids, the resulting construct is in fact birational if the original map is so. A variation of this idea is considered in order to preserve properties of the base ideal, such as Cohen-Macaulayness. Combining the two methods, one is able to produce explicit infinite families of Cohen-Macaulay Cremona maps with prescribed dimension, codimension, and degree.

Section: The Main Resultsmentioning

confidence: 99%

New constructions of Cremona maps

Costa

Mathematical Research Letters

2013

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“…The reason is the large number of variables that the second method requires (for small cases, both algorithms work fine). The second approach is quite interesting from a theoretical point of view as shown in [5].…”

Section: An Integer Programming Methods Via Hilbert Basesmentioning

confidence: 99%

“…One of the peculiarities of the theory is that even if the given monomials are square-free to start with, the inverse map is generally defined by non-square-free monomials. This makes classification in high degrees, if not the structure of the Cremona monomial group itself, a difficult task (see [5,6]). A complete classification of monomial Cremona maps of degree 2 in any number of variables is given in [5].…”

Section: Application To Cremona Mapsmentioning

confidence: 99%

Combinatorics of Cremona monomial maps

Villarreal

2012

Math. Comp.

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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.

“…By the method of [24] (see also [4]), the inverse map is defined by monomials of degree 3 while the source inversion factor turns out to be the monomial x 0 x 1 x 2 2 x 3 . Thus, the latter is an element in I (3) \ I 3 .…”

Section: The Role Of the Inversion Factormentioning

confidence: 99%

A theorem about Cremona maps and symbolic Rees algebras

Costa

Int. J. Algebra Comput.

Ramos

2014

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This work is about the structure of the symbolic Rees algebra of the base ideal of a Cremona map. We give sufficient conditions under which this algebra has the "expected form" in some sense. The main theorem in this regard seemingly covers all previous results on the subject so far. The proof relies heavily on a criterion of birationality and the use of the so-called inversion factor of a Cremona map. One adds a pretty long selection of examples of plane and space Cremona maps tested against the conditions of the theorem, with special emphasis on Cohen-Macaulay base ideals.