New constructions of Cremona maps

Abstract

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.

The Newton complementary dual revisited

“…The notion of a Newton complementary dual to a set of forms having the same degree has been introduced in [1]. In this work we improve on some of the aspects and results of the basic theory as exposed in the latter paper, adding a striking unifying simplification to it.…”

“…One new fundamental step thereon is the crucial role exerted by the Magnus reciprocal involution (x 0 : · · · : x n ) → (1/x 0 : · · · : 1/x n ) which is the Newton dual to the identity map of P n . A useful observation, not entirely perceived in [1] is that taking the Newton dual is nearly as good as evaluating the original forms on the Magnus involution. This in turn yields a uniform procedure to attain other relations.…”

A connection between birational automorphisms of the plane and linear systems of curves

“…In fact, several authors have dealt with this question recently (see e.g. [18,19]). Thus, given d ∈ N, in this paper we are interested in constructing a birational automorphism of the plane S such that deg(S) = d. For this purpose, we construct a pencil of curves V 1 of degree d (where singularities and simple points are known), and using Algorithm for Pencil Parametrization (see Section 2) we determine a linear subsystem V 2 of dimension 1 of the system of adjoint curves to V 1 .…”

Generalized Newton complementary duals of monomial ideals

“…Meanwhile, the Newton complementary duals of monomial ideals were first introduced by Costa and Simis in [6]. There, the dual operation was applied to study the rational maps between the base ideals and the dual ideals.…”

“…Staptx 2 x 5 , x 1 x 6 , x 4 x 7 uq with good moves Now, suppose that the stable ideal I coming from the diagram D λ´µ satisfies(6) StaGpIq " t m k " x k 1 x k 2 | 1 ď k ď g with k 1 ď k 2 u .…”