Person: Gallego Rodrigo, Francisco Javier
Universidad Complutense de Madrid
Faculty / Institute
Álgebra, Geometría y Topología
Now showing 1 - 10 of 32
PublicationDeformations of canonical double covers(Elsevier, 2016) Gallego Rodrigo, Francisco Javier; Gonzalez, M.; Purnaprajna, B.P.In this paper we show that if X is a smooth variety of general type of dimension m≥2 for which its canonical map induces a double cover onto Y, where Y is the projective space, a smooth quadric hypersurface or a smooth projective bundle over P1, embedded by a complete linear series, then the general deformation of the canonical morphism of X is again canonical and induces a double cover. The second part of the article proves the non-existence of canonical double structures on the rational varieties above mentioned. Our results have consequences for the moduli of varieties of general type of arbitrary dimension, since they show that infinitely many moduli spaces of higher dimensional varieties of general type have an entire “hyperelliptic” component. This is in sharp contrast with the case of curves or surfaces of lower Kodaira dimension. PublicationDeformation of canonical morphisms and the moduli of surfaces of general type(Springer-Verlag, 2010-06-05) Gallego Rodrigo, Francisco Javier; González Andrés, Miguel; Purnaprajna, Bangere P.In this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we prove a criterion that determines when a finite map can be deformed to a one-to-one map. We use this criterion to construct new surfaces of general type with birational canonical map, for different c21 and _ (the canonical map of the surfaces we construct is in fact a finite, birational morphism). Our general results enable us to describe some new components of the moduli of surfaces of general type. We also find infinitely many moduli spaces M(x0,0,y) having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree 2 morphism. PublicationClassification of quadruple Galois canonical covers I(American Mathematical Society, 2008-10) Gallego Rodrigo, Francisco Javier; Purnaprajna, Bangere P.In this article we classify quadruple Galois canonical covers of smooth surfaces of minimal degree. The classification shows that they are either non-simple cyclic covers or bi-double covers. If they are bi-double, then they are all fiber products of double covers. We construct examples to show that all the possibilities in the classification do exist. There are implications of this classification that include the existence of families with unbounded geometric genus, in sharp contrast with triple canonical covers, and families with unbounded irregularity, in sharp contrast with canonical covers of all other degrees. Together with the earlier known results on double and triple covers, a pattern emerges that motivates some general questions on the existence of higher degree canonical covers, some of which are answered in this article. PublicationDeformation of finite morphisms and smoothing of ropes(Cambridge University Press, 2008-03-14) Gallego Rodrigo, Francisco Javier; González Andrés, Miguel; Purnaprajna, Bangere P.In this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1 : 1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension. PublicationSome results on rational surfaces and Fano varieties.(Walter de Gruyter & co, 2001) Gallego Rodrigo, Francisco Javier; Purnaprajna, Bangere P.The goal of this article is to study the equations and syzygies of embeddings of rational surfaces and certain Fano varieties. Given a rational surface X and an ample and base-point-free line bundle L on X, we give an optimal numerical criterion for L to satisfy property N-p. This criterion turns out to be a characterization of property N-p if X is anticanonical. We also prove syzygy results for adjunction bundles and a Reider type theorem for higher syzygies PublicationTriple canonical covers of varieties of minimal degree.(Birkhauser Verlag Ag, 2003) Gallego Rodrigo, Francisco Javier; Purnaprajna, Bangere P.; Seshadri, C. S.; Lakshmibai, VenkatramaniIn this article we study pluriregular varieties X of general type with base-point-free canonical bundle whose canonical morphism has degree 3 and maps X onto a variety of minimal degree Y. We carry out our study from two different perspectives. First we study in Section 2 and Section 3 the canonical ring of X describing completely the degrees of its minimal generators. We apply this to the study of the projective normality of the images of the pluricanonical morphisms of X. Our study of the canonical ring of X also shows that, if the dimension of X is greater than or equal to 3, there does not exist a converse to a theorem of M. Green that bounds the degree of the generators of the canonical ring of X. This is in sharp contrast with the situation in dimension 2 where such converse exists, as proved by the authors in a previous work. Second, we study in Section 4, the structure of the canonical morphism of X. We use this to show among other things the nonexistence of some a priori plausible examples of triple canonical covers of varieties of minimal degree. We also characterize the targets of flat canonical covers of varieties of minimal degree. Some of the results of Section 4 are more general and apply to varieties X which are not necessarily regular, and to targets Y that are scrolls which are not of minimal degree. PublicationSmoothable locally non Cohen-Macaulay multiple structures on curves(Springer, 2014-09) Gallego Rodrigo, Francisco Javier; González, Miguel; Purnaprajna, Bangere P.In this article we show that a wide range of multiple structures on curves arise whenever a family of embeddings degenerates to a morphism of degree . One could expect to see, when an embedding degenerates to such a morphism, the appearance of a locally Cohen-Macaulay multiple structure of certain kind (a so-called rope of multiplicity ). We show that this expectation is naive and that locally non Cohen-Macaulay multiple structures also occur in this situation. In seeing this we find out that many multiple structures can be smoothed. When we specialize to the case of double structures we are able to say much more. In particular, we find numerical conditions, in terms of the degree and the arithmetic genus, for the existence of many locally Cohen-Macaulay and non Cohen-Macaulay smoothable double structures. Also, we show that the existence of these double structures is determined, although not uniquely, by the elements of certain space of vector bundle homomorphisms, which are related to the first order infinitesimal deformations of . In many instances, we show that, in order to determine a double structure uniquely, looking merely at a first order deformation of is not enough; one needs to choose also a formal deformation. PublicationDiseño de materiales virtuales y de blended learning para paliar el efecto de ajuste de presencialidad en los grados de la Facultad de Ciencias Matemáticas(2021) Rodríguez Velasco, Gema; Benavent Merchán, María Teresa; Infante del Río, Juan Antonio; Gallego Rodrigo, Francisco Javier; Prieto Yerro, María Ángeles; Sobrón Fernández, María Inés; Barderas Manchado, Gonzalo; Garrido Carballo, Maria Isabel; Luque Martínez, Teresa ElviraInforme del proyecto de innovación docente para el diseño de una metodología b-learning en distintos grupos de las asignaturas de segundo curso del programa de grados de la Facultad de Matemáticas y desarrollo de materiales de autoaprendizaje y prácticas virtuales. PublicationFlavonols mediate root phototropism and growth through regulation of proliferation-to-differentiation transition.(American Society of Plant Biologists, 2016) Silva Navas, J.; Montero Risueno, M.A.; Manzano, C.; Tellez Robles, B.; Navarro Neila, S.; Gallego Rodrigo, Francisco Javier; Del Pozo, J.C.Roots normally grow in darkness, but they may be exposed to light. After perceiving light, roots bend to escape from light (root light avoidance) and reduce their growth. How root light avoidance responses are regulated is not well understood. Here, we show that illumination induces the accumulation of flavonols in Arabidopsis thaliana roots. During root illumination, flavonols rapidly accumulate at the side closer to light in the transition zone. This accumulation promotes asymmetrical cell elongation and causes differential growth between the two sides, leading to root bending. Furthermore, roots illuminated for a long period of time accumulate high levels of flavonols. This high flavonol content decreases both auxin signaling and PLETHORA gradient as well as superoxide radical content, resulting in reduction of cell proliferation. In addition, cytokinin and hydrogen peroxide, which promote root differentiation, induce flavonol accumulation in the root transition zone. As an outcome of prolonged light exposure and flavonol accumulation, root growth is reduced and a different root developmental zonation is established. Finally, we observed that these differentiation-related pathways are required for root light avoidance. We propose that flavonols function as positional signals, integrating hormonal and ROS pathways to regulate root growth direction and rate in response to light.