The main aim of this paper is proof that the reduction of the singularities of a generalized hypersurfaces agrees with a reduction of singularities of its separatrix; which is a generalization of the result presented in [8] by the first two authors. / El objetivo principal de este artículo es demostrar que la reduccióon de singularidades de una hipersupercie generalizada coincide con una reducción de singularidades de su separatriz; el cual es una generalización del resultado presentado en [8] por los dos primeros autores.

Pour toute variété affine W munie d'une opération d'un groupe réductif G, le schéma de Hilbert invariant est un espace de modules qui classifie les sous-schémas fermés de W, stables par l'opération de G, et dont l'algèbre affine est somme directe de G-modules simples avec des multiplicités finies préalablement fixées. Dans cette thèse , on étudie d'abord le schéma de Hilbert invariant, noté H, qui paramètre les sous-schémas fermés GL(V)-stables Z de W=n1 V oplus n2 V^* tels que k[Z] est isomorphe à la représentation régulière de GL(V) comme GL(V)-module. Si dim(V)<3,on montre que H est une variété lisse, et donc que le morphisme de Hilbert-Chow gamma: H -> W//G est une résolution des singularités du quotient W//G. En revanche, si dim(V)=3, on montre que H est singulier. Lorsque dim(V)<3, on décrit H par des équations et aussi comme l'espace total d'un fibré vectoriel homogène au dessus d'un produit de deux grassmanniennes. On se place ensuite dans le cadre symplectique en prenant n1=n2 et en remplaçant W par la fibre en 0 de l'application moment mu: W -> End(V). On considère alors le schéma de Hilbert invariant H' qui paramètre les sous-schémas contenus dans mu^{-1}(0). On montre que H' est toujours réductible, mais que sa composante principale Hp' est lisse lorsque dim(V)<3. Dans ce cas, le morphisme de Hilbert-Chow est une résolution (parfois symplectique) des singularités du quotient mu^{-1}(0)//G. Lorsque dim(V)<3, on décrit Hp' comme l'espace total d'un fibré vectoriel homogène au dessus d'une variété de drapeaux. Enfin, on obtient des résultats similaires lorsque l'on remplace GL(V) par un autre groupe classique (SL(V), SO(V), O(V), Sp(V)) que l'on fait opérer d'abord dans W=nV, puis dans la fibre en 0 de l'application moment. / Let W be an affine variety equipped with an action of a reductive group G. The invariant Hilbert scheme is a moduli space which classifies the G-stable closed subschemes of W such that the affine algebra is the direct sum of simple G-modules with previously fixed finite multiplicities. In this thesis, we first study the invariant Hilbert scheme, denoted H. It parametrizes the GL(V)-stable closed subschemes Z of W=n1 V oplus n2 V^* such that k[Z] is isomorphic to the regular representation of GL(V) as GL(V)-module. If dim(V)<3, we show that H is a smooth variety, so that the Hilbert-Chow morphism gamma: H -> W//G is a resolution of singularities of the quotient W//G. However, if dim(V)=3, we show that H is singular. When dim(V)<3, we describe H by equations and also as the total space of a homogeneous vector bundle over the product of two Grassmannians. Then we consider the symplectic setting by letting n1=n2 and replacing W by the zero fiber of the moment map mu: W -> End(V). We study the invariant Hilbert scheme H' which parametrizes the subschemes included in mu^{-1}(0). We show that H' is always reducible, but that its main component Hp' is smooth if dim(V)<3. In this case, the Hilbert-Chow morphism is a resolution of singularities (sometimes a symplectic one) of the quotient mu^{-1}(0)//G. When dim(V)=3, we describe Hp' as the total space of a homogeneous vector bundle over a flag variety. Finally, we get similar results when we replace GL(V) by some other classical group (SL(V), SO(V), O(V), Sp(V)) acting first on W=nV, then on the zero fiber of the moment map.

Die sogenannten wunderbaren Modelle für Teilraumanordnungen, eingeführt von DeConcini und Procesi, basierend auf den Techniken der Fulton und MacPherson''schen Kompaktifzierung von Konfigurationsräumen, ermöglichen es, eine Fortsetzung von Feynmandistributionen auf die ihnen zugeordneten divergenten Teilräume in kanonischer Weise zu definieren. Dies wurde in der Dissertation von Christoph Bergbauer ausgearbeitet und diese Arbeit führt die dort präsentierten Ideen weiter aus. Im Unterschied formulieren wir die zentralen Begriffe nicht in geometrischer Sprache, sondern mit Hilfe der partiell geordneten Menge der divergenten Subgraphen eines Feynmangraphen. Dieser Ansatz ist inspiriert durch Feichtners Formulierung der wunderbaren Modellkonstruktion aus kombinatorischer Sicht. Diese Betrachtungsweise vereinfacht die Darstellung deutlich und führt zu einem besseren Verständnis der Fortsetzungs- bzw. Renormierungsoperatoren. Darüber hinaus erlaubt sie das Studium der Renormierungsgruppe, d.h. zu untersuchen, wie sich die renormierten Distributionen unter einem Wechsel des Renormierungspunktes verhalten. Wir zeigen, dass eine sogenannte endliche Renormierung sich darstellen läßt als eine Summe von durch die divergenten Subgraphen bestimmten Distributionen. Dies alles unterstreicht den wohlbekannten Fakt, dass perturbative Renormierung zum größten Teil durch die Kombinatorik von Feynmangraphen bestimmt ist und die analytischen Aspekte nur eine untergeordnete Rolle spielen. / The so-called wonderful models of subspace arrangements, developed in by DeConcini and Procesi, based on Fulton and MacPherson''s seminal paper on a compactification of configuration space, serve as a systematic way to resolve the singularities of Feynman distributions and define in this way canonical renormalization operators. In this thesis we continue the work of Bergbauer where wonderful models were introduced to solve the renormalization problem in position space. In contrast to the exposition there, instead of the subspaces in the arrangement of divergent loci we use the poset of divergent subgraphs of a given Feynman graph as the main tool to describe the wonderful construction and the renormalization operators. This is based on a review article by Feichtner where wonderful models were studied from a purely combinatorial viewpoint. The main motivation for this approach is the fact that both, the renormalization process and the model construction, are governed by the combinatorics of this poset. Not only simplifies this the exposition considerably, but it also allows to study the renormalization operators in more detail. Moreover, we explore the renormalization group in this setting, i.e. we study how the renormalized distributions change if one varies the renormalization points. We show that a so-called finite renormalization is expressed as a sum of distributions determined by divergent subgraphs. The bottom line is that - as is well known, at the latest since the discovery of a Hopf algebra structure underlying renormalization - the whole process of perturbative renormalization is governed by the combinatorics of Feynman graphs while the calculus involved plays only a supporting role.

In 2020 Parusiński and Rond proved that every algebraic set X ⊂ R^n is homeomorphic to an algebraic set X’ ⊂ R^n which is described globally (and also locally) by polynomial equations whose coefficients are real algebraic numbers. In general, the following problem was widely open: Open Problem. Is every real algebraic set homeomorphic to a real algebraic set defined by polynomial equations with rational coefficients? The aim of my PhD thesis is to provide classes of real algebraic sets that positively answer to above Open Problem. In Chapter 1 I introduce a new theory of real and complex algebraic geometry over subfields recently developed by Fernando and Ghiloni. In particular, the main notion to outline is the so called R|Q-regularity of points of a Q-algebraic set X ⊂ R^n. This definition suggests a natural notion of a Q-nonsingular Q-algebraic set X ⊂ R^n. The study of Q-nonsingular Q-algebraic sets is the main topic of Chapter 2. Then, in Chapter 3 I introduce Q-algebraic approximation techniques a là Akbulut-King developed in collaboration with Ghiloni and the main consequences we proved, that are, versions ‘over Q’ of the classical and the relative Nash-Tognoli theorems. Last results can be found in in Chapters 3 &amp; 4, respectively. In particular, we obtained a positive answer to above Open Problem in the case of compact nonsingular algebraic sets. Then, after extending ‘over Q’ the Akbulut-King blowing down lemma, we are in position to give a complete positive answer to above Open Problem also in the case of compact algebraic sets with isolated singularities in Chapter 4. After algebraic Alexandroff compactification, we obtained a positive answer also in the case of non-compact algebraic sets with isolated singularities. Other related topics are investigated in Chapter 4 such as the existence of Q-nonsingular Q-algebraic models of Nash manifolds over every real closed field and an answer to the Q-algebrization problem for germs of an isolated algebraic singularity. Appendices A &amp; B contain results on Nash approximation and an evenness criterion for the degree of global smoothings of subanalytic sets, respectively.