##
**Algorithmic equiresolution of deformations of embedded varieties.**
*(English)*
Zbl 1207.14009

The existence of resolution of singularities of an algebraic variety in arbitrary dimension over a field of characteristic zero was proved by Hironaka. Thereafter several authors have given different constructive proofs. Once we have achieved resolution for a variety, it is natural to wonder if it is possible to resolve simultaneously the singularities of a family of varieties. This paper states a theory of simultaneous resolution of singularities for infinitesimal deformations of embedded varieties, that is, families of embedded varieties parametrized by \(S=Spec(A)\) where \(A\) is an artinian ring.

This resolution involves algorithmic resolution of an embedded variety of arbitrary dimension over a field of characteristic zero. More precisely, the author uses a variant of Villamayor algorithm of resolution of singularities given in S. Encinas and O. Villamayor [Prog. Math. 181, 147–227 (2000; Zbl 0969.14007)], with some tools coming from the desingularization given by J. Wlodarczyk [J. Am. Math. Soc. 18, No. 4, 779–822 (2005; Zbl 1084.14018)]. This variant is called VW-algorithm along the paper.

The author reviews all the notions involved in the construction of Villamayor algorithm and adapts all these notions to the case of deformations of embedded varieties. He has done a formidable work to extend all the concepts to suitable conditions over the special fibers.

The paper starts with a revision of Villamayor algorithm of resolution of basic objects (tuples \(B=(W,I,b,E)\) where \(W\) is a smooth variety, \(I\) is a never zero \(W\)-ideal, \(b\) a positive integer number and \(E\) a set of regular hypersurfaces in \(W\) having only normal crossings), making an extension of all notions to the case where \(\mathcal{A}\) is a collection of artinian local rings \((A,M)\) such that the residue field \(k=A/M\) has characteristic zero. So he works in the context of \(A\)-basic objects, that is a basic object over a ring \(A\in \mathcal{A}\). One of the key points is the definition of the adapted hypersurfaces playing the analogous role to the hypersurfaces of maximal contact, and the proof of that this notion is stable under permissible transformation. The step that differs from Villamayor algorithm is the use of the homogenized ideal, due to Wlodarczyk, instead of the generalized basic objects to solve the problem of the patching when there are many adapted hypersurfaces. This is performed passing from the \(A\)-basic object \(B\) to its homogenized \(\mathcal{H}(B)\) to make induction on the dimension of the ambient space.

He proves that the algorithmic equiresolution of \(A\)-basic object leads to the algorithmic equiprincipalization of triples \((W\rightarrow S,I,E)\) over \(A\in\mathcal{A}\), and hence resolution of embedded varieties over an Artin ring \(A\in \mathcal{A}\). In this case it is said that the relative embedded \(A\)-variety is algorithmically equisolvable.

The author also includes several interesting examples along the article, such as the example of an \(A\)-basic object that is not algorithmically equisolvable.

The article is self contained, the author includes the necessary theoretical background and an appendix of revision of useful results that not always appear in the literature.

This resolution involves algorithmic resolution of an embedded variety of arbitrary dimension over a field of characteristic zero. More precisely, the author uses a variant of Villamayor algorithm of resolution of singularities given in S. Encinas and O. Villamayor [Prog. Math. 181, 147–227 (2000; Zbl 0969.14007)], with some tools coming from the desingularization given by J. Wlodarczyk [J. Am. Math. Soc. 18, No. 4, 779–822 (2005; Zbl 1084.14018)]. This variant is called VW-algorithm along the paper.

The author reviews all the notions involved in the construction of Villamayor algorithm and adapts all these notions to the case of deformations of embedded varieties. He has done a formidable work to extend all the concepts to suitable conditions over the special fibers.

The paper starts with a revision of Villamayor algorithm of resolution of basic objects (tuples \(B=(W,I,b,E)\) where \(W\) is a smooth variety, \(I\) is a never zero \(W\)-ideal, \(b\) a positive integer number and \(E\) a set of regular hypersurfaces in \(W\) having only normal crossings), making an extension of all notions to the case where \(\mathcal{A}\) is a collection of artinian local rings \((A,M)\) such that the residue field \(k=A/M\) has characteristic zero. So he works in the context of \(A\)-basic objects, that is a basic object over a ring \(A\in \mathcal{A}\). One of the key points is the definition of the adapted hypersurfaces playing the analogous role to the hypersurfaces of maximal contact, and the proof of that this notion is stable under permissible transformation. The step that differs from Villamayor algorithm is the use of the homogenized ideal, due to Wlodarczyk, instead of the generalized basic objects to solve the problem of the patching when there are many adapted hypersurfaces. This is performed passing from the \(A\)-basic object \(B\) to its homogenized \(\mathcal{H}(B)\) to make induction on the dimension of the ambient space.

He proves that the algorithmic equiresolution of \(A\)-basic object leads to the algorithmic equiprincipalization of triples \((W\rightarrow S,I,E)\) over \(A\in\mathcal{A}\), and hence resolution of embedded varieties over an Artin ring \(A\in \mathcal{A}\). In this case it is said that the relative embedded \(A\)-variety is algorithmically equisolvable.

The author also includes several interesting examples along the article, such as the example of an \(A\)-basic object that is not algorithmically equisolvable.

The article is self contained, the author includes the necessary theoretical background and an appendix of revision of useful results that not always appear in the literature.

Reviewer: Rocío Blanco (Cuenca)

### MSC:

14B05 | Singularities in algebraic geometry |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

14E99 | Birational geometry |

14D99 | Families, fibrations in algebraic geometry |

14H20 | Singularities of curves, local rings |

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\textit{A. Nobile}, Rev. Mat. Iberoam. 25, No. 3, 995--1054 (2009; Zbl 1207.14009)

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