Augmentation of singularities, quasi-homogeneity and good real images of map-germs

• Autores: Ignacio Breva Ribes
• Directores de la Tesis: Raúl Oset Sinha (dir. tes.)
• Lectura: En la Universitat de València ( España ) en 2026
• Idioma: inglés
• Número de páginas: 204
• Tribunal Calificador de la Tesis: Farid Tari (presid.) , Juan José Nuño Ballesteros (secret.) , Maria Elenice Rodrigues Hernandes (voc.)
• Programa de doctorado: Doctor en Programa Oficial de Posgrado en matemáticas
• Menciones:
  • Internacional
• Calificación: Sobresaliente Cum Laude
• Enlaces
  • Tesis en acceso abierto en:  TESEO  RODERIC 
• Resumen

  • We study how the process of augmentation of singularities of map-germs preserves certain properties of the augmented map-germ and its augmenting function. In particular, we give sufficient criteria to determine when the simplicity of the latter two implies the simplicity of the augmentation, and we give two upper bounds on the codimension of the augmentation. One of these bounds allows us to prove the Mond conjecture for augmentations of map-germs which satisfy the Mond conjecture. The other one has a concrete interpretation for map-germs with source dimension greater than target dimension; we also discuss its possible interpretation in other cases. We give a characterization of when a map-germ is in the $\mathscr A$-class of an augmentation, correcting a previous result by Houston. We also characterize simplicity in the case that the augmented map-germ has codimension 1 (in the left-right equivalence sense) or that the augmenting function is a Morse singularity. The problem of whether the $\mathscr A$-class of an augmentation depends on the choice of $1$-parameter stable unfolding is also tackled via two methods, one using an adaptation of $\phi$-equivalence and via substantiality of the unfoldings, and another more direct method comparing the $\mathscr A_e$-normal spaces of augmentations. We conclude that when augmenting by quasi-homogeneous functions or by substantial unfoldings the choice of stable unfolding is irrelevant. This allows us to describe a full classification of simple augmentations of equidimensional map-germs in dimension 4.

    We also take Houston#s concept of substantial 1-parameter stable unfoldings and generalize it to any number of parameters, showing that it is a necessary condition for a map-germ to be quasi-homogeneous. This is applied to concrete examples, to discard quasi-homogeneity.

    The converse implication, is showed for equidimensional map-germs with minimal stable unfolding or of multiplicity 3. We conjecture that substantiality characterizes quasi-homogeneity in general.

    Finally, we give a necessary condition on when a map-germ admits a good real image for finitely determined map-germs with source dimension smaller than the target dimension. Concretely, we show that for corank $1$ map-germs admitting a good real image, the nonempty, positive dimensional multiple point spaces must have Milnor number at most $1$. This condition gives clues on how to obtain such a perturbation if it exists. We show this with examples and provide context on how a classification of jets of map-germs can be obtained, classifying all simple map-germs with good real perturbation from $(\bbc^3,0)$ to $(\bbc^4,0)$. A conjecture by Cooper and Mond is also solved, showing that the homotopy type of a good real perturbation is the same as that of its complexification, for map-germs parametrizing hypersurfaces. The proofs for this last part rely heavily on results from Smith theory, of which we give a description in the annex.