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Orlando E. Villamayor

Publications and source records attributed to Orlando E. Villamayor.

5 recordsLinked to original sources
arXiv2024

Differentiably simple rings and ring extensions defined by $p$-basis

We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit $p$-basis. These extensions are called "Galois extensions of exponent one". For such an extension $A\subset C$, we introduce an $A$-scheme, called the "Yuan scheme", which parametrizes subextensions $A\subset B\subset C$ such that $B\subset C$ is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers.

math.AC↗
arXiv2018

Finite morphisms and simultaneous reduction of the multiplicity

Let $X$ be a singular algebraic variety defined over a field $k$, with quotient field $K(X)$. Let $s \geq 2$ be the highest multiplicity of $X$ and $F_s(X)$ the set of points of multiplicity $s$. If $Y\subset F_s(X)$ is a regular center and $X\leftarrow X_1$ is the blow up at $Y$, then the highest multiplicity of $X_1$ is less than or equal to $s$. A sequence of blow ups at regular centers $Y_i \subset F_s(X_i)$, say $X \leftarrow X_1 \leftarrow \dotsb \leftarrow X_n$, is said to be a {\em simplification} of the multiplicity if the maximum multiplicity of $X_n$ is strictly lower than that of $X$, that is, if $F_s(X_n) $ is empty. In characteristic zero there is an algorithm which assigns to each $X$ a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties $β: X'\to X$ of generic rank $r \geq 1$ (i.e., $[K(X'):K(X)]=r$). We will see that, when imposing suitable conditions on $β$, there is a strong link between the strata of maximum multiplicity of $X$ and $X'$, say $F_{s}(X)$ and $F_{rs}(X')$ respectively. In such case, we will say that the morphism is strongly transversal. When $β: X'\to X$ is strongly transversal one can obtain information about the simplification of the multiplicity of $X$ from that of $X'$ and vice versa. Finally, we will see that given a singular variety $X$ and a finite field extension $L$ of $K(X)$ of rank $r \geq 1$, one can construct (at least locally, in étale topology) a strongly transversal morphism $β: X'\to X$, where $X'$ has quotient field $L$.

math.AG↗
arXiv2011

Monoidal transforms and invariants of singularities in positive characteristic

The problem of resolution of singularities in positive characteristic can be reformulated as follows: Fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, such an $n$-sequence is defined in two steps: the first consisting in the transformation of $X$ to a hypersurface with $n$-fold points in the so called monomial case. The second step consists in the elimination of these $n$-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to define a notion of strong monomial case which parallels that of monomial case in characteristic zero: If a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.

math.AG↗
arXiv2011

On elimination of variables in the study of singularities in positive characteristic

The objective of this paper is to discuss invariants of singularities of algebraic schemes over fields of positive characteristic, and to show how they yield the simplification of singularities. We focus here on invariants which arise in an inductive manner, namely by successive elimination of variables. When applied to hypersurface singularities they lead us to a refinement of the notion of multiplicity. The main theorem proves that, under some numerical conditions expressed by these invariants, singularities can be simplified by blowups at centers prescribed by this refinement.

math.AG↗