arXiv subjects
Publications and source records attributed to Orlando Villamayor.
A sequence of monoidal transformations is defined, in terms of invariants, for a singular hypersurface embedded in a smooth scheme of positive characteristic. Some examples are added to illustrate the improvement of singularities by this procedure.
We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, also known as a reduction to the monomial case.
The paper is motivated on the open problem of resolution of singularities in positive characteristic. The aim is to present a form of induction which is different from that used by Hironaka. In characteristic zero induction is formulated by restriction to smooth hypersurfaces (hypersurfaces of maximal contact). Our alternative approach, introduced here, replaces restrictions to smooth sub-schemes by generic projections on smooth schemes of smaller dimension. We also introduce a generalization of the discriminant, and our result makes use of the elimination theory. In the case of fields of characteristic zero, elimination gives exactly the same information as the form of induction used by Hironaka.
Let $V$ be a smooth scheme over a field $k$, and let $\{I_n, n\geq 0\}$ be a filtration of sheaves of ideals in $\calo_V$, such that $I_0=\calo_V$, and $I_s\cdot I_t\subset I_{s+t}$. In such case $\bigoplus I_n$ is called a Rees algebra. A Rees algebra is said to be a Diff-algebra if, for any two integers $N>n$ and any differential operator $D$ of order $n$, $D(I_N)\subset I_{N-n}$. Any Rees algebra extends to a smallest Diff-algebra. There are two ways to define extensions of Rees algebras, and both are of interest in singularity theory. One is that defined by taking integral closures (in which a Rees algebra is included in its integral closure), and another extension is that defined, as above, in which the algebra is extended to a Diff-algebra. Surprisingly enough, both forms of extension are compatible in a natural way. Namely, there is a compatibility of higher differential operators with integral closure which we explore here under the assumption that $V$ is smooth over a perfect field.
Algebraic varieties which are locally isomorphic to open subsets of affine space will be called {\em plain}. Plain varieties are smooth and rational. The converse is true for curves and surfaces, and unknown in general. It is shown that plain varieties are stable under blowup in smooth centers.
We present an application of elimination theory to the study of singularities over arbitrary fields, particularly to the open problem of resolution. A partial extension of a function, defining resolution of singularities over fields of characteristic zero, is discussed here in positive characteristic.
Embedded principalization of ideals in smooth schemes, also known as Log-resolutions of ideals, play a central role in algebraic geometry. If two sheaves of ideals, say $I_1$ and $I_2$, over a smooth scheme $V$ have the same integral closure, it is well known that Log-resolution of one of them induces a Log-resolution of the other. On the other hand, in case $V$ is smooth over a field of characteristic zero, an algorithm of desingularization provides, for each sheaf of ideals, a unique Log-resolution. In this paper we show that algorithms of desingularization define the same Log-resolution for two ideals having the same integral closure. We prove this result here by using the form of induction introduced by Włodarczyk. We extend the notion of Log-resolution of ideals over a smooth scheme $V$, to that of Rees algebras over $V$; and then we show that two Rees algebras with the same integral closure undergo the same constructive resolution. The key point is the interplay of integral closure with differential operators.
Let $X$ be a closed subscheme embedded in a scheme $W$ smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I}(X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $π: W_r\longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r\subset W_r$ denotes the strict transform of $X\subset W$ then: 1) The morphism $π:W_r\longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem); 2) The {\em total transform} of ${\mathcal I}(X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (i.e. ${\mathcal I}(X){\mathcal O}_{W_r}={\mathcal L}\cdot{\mathcal I}(X_r)$). This result is stronger than Hironaka's Theorem, in fact (2) is novel and does not hold for desingularizations which follow Hironaka's line of proof unless $X$ is a hypersurface. We will say that $W_r\longrightarrow W$ defines a {\em Strong Desingularization of $X$}.