arXiv subjects

Augusto Nobile

Publications and source records attributed to Augusto Nobile.

4 recordsLinked to original sources

arXiv2018

On the Nash problem for surfaces in positive characteristic

This paper seeks to prove the bijectivity of the "Nash mapping" from the set of irreducible components of the scheme parametrizing analytic arcs on an algebraic surface $X$ whose origin is a singular point, into the set of irreducible components of the exceptional locus of a minimal desingularization $X'$ of $X$ when the base field has positive characteristic. The idea is to view the surface as a specialization of another defined over a field of characteristic zero. A number of related results are proved. Among them, the construction of a scheme of arcs for a suitable one parameter family of surfaces which, by using a theorem of M. Artin on lifting of normal surface singularities to characteristic zero, seems a reasonable candidate to be the desired tool. But there are some points, necessary for a complete proof, which are not verified yet.

math.AG↗

arXiv2013

Resolution Algorithms and Deformations

An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consiting of a sheaf of ideals and a positive integer. This approach is particularly simple and, as indicated by some results shown here, it seems suitable for applications to a good theory of simultaneous algorithmic resolution of singularities, specially for families parametrized by the spectrum of an artinian ring.

math.AG↗

arXiv2010

Algorithmic equiresolution of possibly non-reduced families of singularities

This paper studies the concept of algorithmic equiresolution of a family of embedded varieties or ideals, which means a simultaneous resolution of such a family compatible with a given (suitable) algorithm of resolution in characteristic zero. The paper's approach is more indirect: it primarily considers the more general case of families of basic objects (or marked ideals). A definition of algorithmic equiresolution is proposed, which applies to families whose parameter space T may be non-reduced, e.g., the spectrum of a suitable artinian ring. Other definitions of algorithmic equiresolution are also discussed. These are geometrically very natural, but the parameter space T of the family must be assumed regular. It is proven that when T is regular all the proposed definitions are equivalent.

math.AG↗

arXiv2009

Algorithmic equiresolution of deformations of embedded varieties

A theory of simultaneous resolution of singularities for families of embedded varieties (over a field of characteristic zero) parametrized by the spectrum of a suitable artinian ring, and compatible with a given algorithm of resolution, is presented. As usually, this a simple consequence of a similar theory for analogous families of basic objects, to which the main portion of this article is devoted.

math.AG↗