Invariantes de singularidades em característica positiva

Abstract

This dissertation mainly follows the works of \cite{Gr2} and \cite{Gr5}.

Considering $\mathbb{K}$ a field, algebraically closed with positive characteristic, we presented invariants of singularities in $\mathbb{K}[[\underline{x}]],$ the local $\mathbb{K}$- algebra of the formal power series, such as Milnor and Tjurina numbers.

Two equivalence relations are defined on $\mathbb{K}[[\underline{x}]]$, right equivalence and contact equivalence. The concept of finite determinancy of $f\in\mathbb{K}[[\underline{x}]]$ is defined with respect to those equivalence relations, the finite determinancy is also expressed in terms of the Milnor and Tjurina numbers.

We show that a necessary condition for $f\!\in\!\mathbb{K}[[\underline{x}]]$ to be finitely determined by the right (respectively contact) is that it has an isolated singularity (respectively is a hypersurface with isolated singularity); the necessary condition is based on a technical lemma considering $\mathbb{K}[[\underline{x}]]$ with the $\mathfrak{m}$-adic topology. Finally, considering that the orbit application, in general, is not separable in positive characteristic, it is proved that the condition is also sufficient.