RT info:eu-repo/semantics/doctoralThesis
T1 Foliaciones de codimensión uno Newton no degeneradas
A1 Molina Samper, Beatriz
A2 Universidad de Valladolid. Facultad de Ciencias
K1 Foliaciones (Matemáticas)
K1 Superficies (Matemáticas)
AB The main topic of this research is the study of “Newton non-degeneratecodimension one foliations”.The non-degenerate singularities for hypersurfaces have been described classically by A.Kouchnirenko in [35]; let us give a quick description of them.We consider a germ of hypersurfacein (Cn, 0), defined locally by a reduced equation f = 0 in local coordinates z = (z1, z2, . . . , zn).We take the Taylor’s expansion of f, we consider the convex hull of the   2 Rn 0 such that   6= 0 and we add to it the first orthant Rn 0. In this way it is obtained the Newton polyhedron1of f. We consider its compact boundary   and we say that a singularity is “non-degenerate”if the coefficients are “generic” in a sense that we will define later. This class of singularities isopen and dense in the space of coefficients when   is fixed. Also M. Oka does a study in [36]of the non-degenerate singularities for the case of complete intersections.Taking a logarithmic point of view, we can define a Newton polyhedron associated to a germof differential form or vector field, once we fix a system of coordinates. From a more geometricalapproach, the fact of considering a normal crossings divisor in the ambient space determines thecoordinates we are going to consider. On this way, we can define not just a single polyhedron,but a whole polyhedra system, each one associated to one of the strata naturally given by thedivisor as we will see in Chapter 2.A foliated space consists of a codimension one foliation F in a complex analytic space M, togetherwith a normal crossings divisor E   M. Most of the definitions, properties and results wepresent in this work concerns the foliated space (M,E,F) and not just to the foliation F. In thegeneral theory established in Chapter 4, we introduce the concept of “Newton non-degeneratefoliated space” which, of course, coincides with the classical one for germs of hypersurfaces,when we consider germs of foliations having a holomorphic first integral. On the other hand,once we have a normal crossings divisor in the ambient space, we can talk about “combinatorialblowing-ups”. They are blowing-ups centered at the closure of one of the strata determined bythe divisor. We extend the definition introduced by M.I.T. Camacho and F. Cano in [9] and wesay that a codimension one foliation is of “toric type” if we obtain only “simple points” aftera combinatorial sequence of blowing-ups, that is, if it has a “combinatorial desingularization”.
YR 2019
FD 2019
LK http://uvadoc.uva.es/handle/10324/39469
UL http://uvadoc.uva.es/handle/10324/39469
LA eng
NO Departamento de Algebra, Geometría y Topología
DS UVaDOC
RD 25-nov-2024