Good points and constructive resolution of singluarities

1998/01/01 by S. Encinas, Orlando E. Villamayor · 6 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Polynomial and algebraic computation #Resolution of singularities #Mathematics #Base (topology) #Resolution (logic) #Field (mathematics) #Constructive #Class (philosophy) #Zero (linguistics) #Function (biology) #Gravitational singularity #Base change #Discrete mathematics #Function field #Combinatorics #Point (geometry) #Pure mathematics #Geometry #Mathematical analysis #Computer science #Artificial intelligence

paper · pdf · doi:10.1007/bf02392749

openalex publication_date 1998/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11

Abstract

In this paper we present a new algorithm of resolution of singularities over fields of characteristic zero, making use of invariants that come from Abhyankar's good point theory [Abl]. We also prove new properties on constructive (or algorithmic) desingularization. Let us explain what we mean by an algorithm of resolution. Consider a pair (X, W) where W is a regular variety over a base field k (of characteristic zero) not necessarily irreducible (i.e. W smooth over k), and XC_W is a closed non-empty subscheme. Call C the class of all such pairs (over different base fields k). Natural maps (~ I (X) ,W1) ~' ) (X,W) arise within this class, for instance if ~: W1--,W is a smooth map over a fixed field k, or if ~: W1--*W arises from an arbitrary change of base field. Fix now a totally ordered set (I, ~<) and suppose assigned, for each pair 7~= (X, W) of U, a function ~p: X--+I which is upper-semi-continuous and takes only finitely many values, say ~1, ,.., C~rC--I. Let max~bp be the biggest ai and set

Citations

• Smoothness, Semistability, and Toroidal Geometry
• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I
• Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant
• Smoothness, semi-stability and alterations

Cited by

• Strong resolution of singularities in characteristic zero
• Simple Hironaka resolution in characteristic zero
• The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)
• On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for)
• Resolution of singularities and geometric proofsof the Łojasiewicz inequalities
• Resolution of singularities [wikipedia]