Integral Closure, Whitney Stratification, and Bi-Lipschitz Invariants on Toric Varieties

Abstract

This work, we investigate the interation between combinatorial structures, algebraic and geometric properties, and metric aspects in Singularity Theory, with affine toric varieties as the main setting. The central objective is to develop an approach that allows one to describe invariants and properties of singularities using combinatorial data associated with semigroups and Newton polyhedra. First, we study the integral closure of non-degenerate ideals in rings of analytic functions associated with affine toric varieties. We show that, under suitable non-degeneracy conditions with respect to the Newton polyhedron, the integral closure of such ideals admits a precise combinatorial description, extending classical results known in the regular case. Next, we investigate toric surfaces in C3 and construct a Whitney stratification adapted to their combinatorial structure. We prove that, under appropriate conditions, it is possible to obtain a stratification with a reduced number of strata while preserving Whitney regularity conditions. Finally, we analyze the bi-Lipschitz invariance of local invariants of singularities. We establish algebraic conditions ensuring the invariance of the Łojasiewicz exponent and the local Euler obstruction. The obtained results show that toric varieties constitute a particularly suitable setting to translate analytic, algebraic, and metric phenomena into combinatorial terms, providing new tools for the systematic study of singularities in more general contexts.