Combinatorics of Cremona monomial maps

Advances in Mathematics

2012

117

One develops ab initio the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A numerical invariant of a rational map is introduced, called the Jacobian dual rank. It is proved that a rational map in this general setup is birational if and only if the Jacobian dual rank attains its maximal possible value. Even in the "classical" case where the source variety is irreducible there is some gain for this invariant over the degree of the map as it is, on one hand, intrinsically related to natural constructions in commutative algebra and, on the other hand, is effectively straightforwardly computable. Applications are given to results so far only known in characteristic zero. In particular, the surprising result of Dolgachev concerning the degree of a plane polar Cremona map is given an alternative conceptual angle.The simplification comes about by showing that birationality is controlled by the behavior of a unique numerical invariant -called the Jacobian dual rank of a rational map. Alas, this sounds like old knowledge because the classical theory also depends only on the degree of the rational map. However, the latter is in full control only in the integral case. Habitually, in positive characteristic one treats the inseparability degree apart from the main stream of the natural ideas in birational theory. The new invariant introduced here looks more intrinsic and makes no explicit reference to inseparability, so the criterion itself and the applications will be characteristic-free. Finally, the Jacobian dual rank is straightforwardly effectively computable in the usual implementation of the Gröbner basis algorithm, an appreciable advantage over the field degree.In addition, the Jacobian dual rank calls attention to several aspects of the theory of Rees algebras and base ideals of maps, a trend sufficiently shown in many modern accounts (see, e.g., [2], [5], [6], [7]).The paper is divided in two sections. The first section hinges on the needed background to state the general criterion of birationality.In the initial subsections we develop the ground material on rational and birational maps on a reduced source. Our approach is entirely algebraic, but we mention the transcription to the geometric side. A degree of care is required to show that the present notion is stable under the expected manipulations from the "classical" case. One main result in this part is Proposition 1.11 which drives us back to an analogue of the field extension version.The main core is the subsequent subsection, where we introduce the Jacobian dual rank and prove the basic characteristic-free criterion of birationality in terms of this rank. The criterion also holds component-wise as possibly predictable (but not obviously proved!). We took pains to transcribe the criterion into purely geometric terms, except for the Jacobian dual rank itself, whose geometric meaning is not entirely apparent at this stage. This concept has evolved continuously from previous notio...

“…Often this can be computed by a careful scrutiny of the Hilbert series (see, e.g. [8, Proposition 3.1]) or by local structure (see [8,Examples 2.4,2.5,] and also [17,Theorem 3.1]).…”

Section: It Remains To Prove (I)mentioning

confidence: 99%

A characteristic-free criterion of birationality

Dória

Hassanzadeh

Advances in Mathematics

2012

117

“…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

Self Cite

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.

“…We will also make heavy use throughout of the main result in [16] giving the integer arithmetic counterpart of a monomial Cremona situation:…”

Section: Terminology and Basic Combinatorial Criterionmentioning

confidence: 99%

“…Since the inverse has to be given by monomials as well, as shown in ibid., the question made sense. This question was eventually solved in [16] a few years later.…”

Section: Introductionmentioning

confidence: 99%

“…Though the criteria themselves have an expected simplicity and afford effective computation (see [16,Section 4]) -although facets of the computation are close to NP-hard problems -the practical use in theoretical classification is by no means obvious, often requiring quite a bit of ingenuity. Having traded geometrical tools by integer linear algebra one pays a price in that further precision has to be exercised.…”

Section: Introductionmentioning

confidence: 99%

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Cremona maps defined by monomials

Costa

Journal of Pure and Applied Algebra

2012

Self Cite

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter suggests that facets of monomial Cremona theory may be NP-hard.