Friday, March 13

on-line on zoom

• 14:00-14:50 Thaís Maria Dalbelo (Universidade Federal de São Carlos) Whitney equisingularity for families of hypersurfaces in toric varieties

  Abstract: In this work, we establish conditions under which a family \(\{f_t\}\) of functions, possibly with non-isolated singularities, defined on a toric variety gives rise to a Whitney equisingular family of hypersurfaces \(\{f_t^{−1}(0)\}\). In a recent and notable work, Eyral and Oka provided an important framework for studying Whitney equisingularity in families with non-isolated singularities in \(\mathbb{C}^n\). Our approach builds directly on their concepts, such as admissibility, local tameness, and the role of Newton boundaries, by extending them to the toric setting. However, because toric varieties may present arbitrary singular sets, these extensions alone are not sufficient. To overcome this, we combine their framework with fundamental properties of Whitney stratifications and finite morphisms, thereby extending the classical theory to the broader context of toric varieties. Moreover, when \(\mathbb{C}^n\) is viewed as a toric variety, our conditions recover exactly those obtained by Eyral and Oka.
  
  This is a joint work with Danilo da Nóbrega Santos

• 15:00-15:50 Edson Sampaio (Universidade Federal do Ceará) On the Milnor fibres of initial forms of topologically equivalent holomorphic functions

  Abstract: Budur, Fernández de Bobadilla, Le and Nguyen (2022) conjectured that if two germs of holomorphic functions are topologically equivalent, then the Milnor fibres of their initial forms are homotopy equivalent. In this talk, we will give affirmative answers to this conjecture, e.g., it is true in the case of plane curves. We will show also that a positive answer to this conjecture implies in a positive answer to the famous Zariski multiplicity conjecture both in the case of right equivalence or in the case of hypersurfaces with isolated singularities.

• 16:00-16:50 Maria Michalska (ICMC USP) Nullstellensatz in families of arc-regular functions

  Astract: We present a general approach to noetherianity of topologies in families of functions that do not admit bump functions. This allows us to give a simple uniform proof of Nullstellensatz by its equivalence with Lojasiewicz inequality. This characterization also provides examples of families of definable arc-regular functions that do not admit a Nullstellensatz.