New Ideas for Resolution of Singularities in Arbitrary Characteristic

TL;DR

This paper introduces a novel approach using toric geometry and Newton polyhedra to improve singularities in algebraic varieties over fields of any characteristic, advancing toward a general resolution of singularities.

Contribution

It develops a new method involving upward subdivisions of Newton polyhedra and associated toric morphisms to systematically improve singularities in arbitrary characteristic.

Findings

The method guarantees a decrease in a numerical invariant measuring singularity severity.

It demonstrates that the proposed approach can always lead to local uniformization.

The results establish a foundation for resolving singularities in arbitrary characteristic and dimension.

Abstract

Let $k$ be \emph{any} algebraically closed field in any characteristic, let $R$ be any regular local ring such that $R$ contains $k$ as a subring, the residue field of $R$ is isomorphic to $k$ as $k$-algebras and $\dim R\geq 1$, let $P$ be any parameter system of $R$ and let $z\in P$. We consider any $\phi\in R$ with $\phi\neq 0$. In our main theorem we assume several conditions depending on $P$, $z$ and Newton polyhedrons. By our assumptions the normal fan $\Sigma$ of the Newton polyhedron $\Gamma_+(P,\phi)$ of $\phi$ over $P$ has simple structure and we can make a special regular subdivision $\Sigma^*$ of $\Sigma$ called an upward subdivision, starting from a regular cone with dimension equal to $\dim R$ and repeating star subdivisions with center in a regular cone of dimension two. Let $X$ and $\sigma:X\rightarrow\Spec(R)$ denote the toric variety over $\Spec(R)$ and the toric morphism associated with $\Sigma^*$. We consider any closed point $a\in X$ such that $\sigma(a)$ is the unique closed point of $\Spec(R)$ and the morphism $\sigma^*: R \rightarrow\mathcal{O}_{X,a}$ of local $k$-algebras induced by $\sigma$. We show that our numerical invariant of $\sigma^*(\phi)\in \mathcal{O}_{X,a}$ measuring the badness of the singularity is strictly less than the same invariant of $\phi\in R$ and the singularity $\phi$ is strictly improved by $\sigma$. We notice that this result opens a way toward the theory of resolution of singularities in arbitrary characteristic. We add several submain theorems to make bridges toward it and to show that our assumptions of the main theorem are not strong. By these results we can show that in a mathematical game with two players A and B related to the resolution of singularities of $\phi$, the player A can always win the game after finite steps. It follows "the local uniformization theorem in arbitrary characteristic and in arbitrary dimension".