Low codimension Fano–Enriques threefolds

In this paper we study Fano threefolds with a torsion divisor (Fano–Enriques). Due to this torsion divisor, they can be described as quotients of Fano threefolds by a finite abelian group action. We start from lists of Fano threefolds by Reid, Fletcher and Altınok and check which of them admit such an action with a Fano–Enriques quotient

A Maple package to deal with the birationality of curves and surfaces parametrizations

Maple Conference 2020. Communications in Computer and Information Science book series (CCIS volume 1414)In this paper we present the Maple package Luroth for dealing with the birationality of curves and surfaces parametrizations. The procedures in the package decide whether a given, either curve or surface, parametrization is injective by computing its degree map. In addition, if the parametrization is not injective, it determines a birational reparametrization. For the curve case, the corresponding command always provides an optional answer. For the surface case, not all cases are covered. Nevertheless, we illustrate using Maple some new ideas on how to approach those surface cases not covered in the package.Agencia Estatal de Investigació

A canonical form for the continuous piecewise polynomial functions

We present in this paper a canonical form for the elements in the ring of continuous piecewise polynomial functions. This new representation is based on the use of a particular class of functions fCi(P) : P 2 Q[x]; i = 0; : : : ; deg(P)g de ned by Ci(P)(x) = 0 if x P(x) if x where is the i-th real root of the polynomial P. These functions will allow us to represent and manipulate easily every continuous piecewise polynomial function through the use of the corresponding canonical form. It will be also shown how to produce a \rational" representation of each function Ci(P) allowing its evaluation by performing only operations in Q and avoiding the use of any real algebraic number. Keywords: Continuous piecewise polynomial functions; Pierce-Birkho conjecture; Canonical form for func- tions; Conversion algorithms

Covering rational surfaces with rational parametrization images

Let S be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps f , g, h : A2 S ⊂ P n such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps f , g˜ : A2 S, such that the union of its images covers the affine surface S ∩ An . In the affine case, the number of rational maps involved in the cover is in general optimal.Ministerio de Ciencia, Innovación y UniversidadesJ. Caravantes, J.R. Sendra and C. Villarino belong to the Research Group ASYNACS (Ref. CT-CE2019/683). D. Sevilla is a member of the research group GADAC and is partially supported by Junta de Extremadura and FEDER funds (group FQM024)

On the existence of birational surjective parametrizations of affine surfaces

In this paper we show that not all affine rational complex surfaces can be parametrized birational and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary condition for S to admit a birational surjective parametrization from an open subset of the affine complex plane is that the curve at infinity of S must contain at least one rational component. As a consequence of this result we provide examples of affine rational surfaces that do not admit birational surjective parametrizations.Ministerio de Economía y CompetitividadJunta de ExtremaduraEuropean Regional Development Fun

Transforming ODEs and PDEs from radical coefficients to rational coefficients

We present an algorithm that transforms, if possible, a given ODE or PDE with radical function coefficients into one with rational coefficients by means of a rational change of variables so that solutions correspond one-to-one. Our method also applies to systems of linear ODEs. It is based on previous work on reparametrization of radical algebraic varieties.Agencia Estatal de InvestigaciónUniversidad de AlcaláJunta de Extremadur

A note about rational surfaces as unions of affine planes

J. Caravantes and C. Villarino belong to the Research Group ASYNACS (Ref. CT-CE2019/683). D. Sevilla is a member of the research group GADAC and is partially supported by Junta de Extremadura and Fondo Europeo de Desarrollo Regional (GR21055)Demostramos que cualquier superficie proyectiva racional lisa, sobre el cuerpo de los números complejos, admite un recubrimiento por medio de tres subconjuntos isomorfos a planos afines.We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.Agencia Estatal de Investigació

Sufficient conditions for the surjectivity of radical curve parametrizations

J. Caravantes and C. Villarino belong to the Research Group ASYNACS (Ref. CT-CE2019/683). D. Sevilla is a member of the research group GADAC and is partially supported by Junta de Extremadura and Fondo Europeo de Desarrollo Regional (GR21055).En este artículo se introduce la noción de parametrización radical suprayectiva y se dan condiciones suficientes para que una parametrización radical de una curva sea suprayectiva.In this paper, we introduce the notion of surjective radical parametrization and we prove sufficient conditions for a radical curve parametrization to be surjective.Agencia Estatal de Investigació

On the implicit equation of conics and quadrics offsets

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in ?A Treatise on the Analytic Geometry of Three Dimensions? by G. Salmon for offsets to non-degenerate conics and central quadrics.This research was funded by the Spanish Ministerio de Economía y Competitividad and by the European Regional Development Fund (ERDF), under the project MTM2017-88796-P

Antena margarita

A new antenna concept, which is named by us as “DAISY ANTENNA”, is presented in this communication. The antenna concept is based on a circular array of wire radiating elements, and its radiating field behaviour is determined by the current distribution in wire elements. This configuration provides a linear polarised field with a maximum of radiation pattern in the perpendicular direction to the daisy surface. This special array concept can be also applied to other kind of linear polarisation radiating elements, as for example to microstrip patches or waveguide horns, and consequently, the antenna directivity can be improved