## 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.

### 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

### Citations:

Zbl 0969.14007; Zbl 1084.14018
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### References:

 [1] André, M.: Homologie des algèbres commutatives. Die Grundlehren der mathematischen Wissenschaften 206 . Springer-Verlag, Berlin-New York, 1974. · Zbl 0284.18009 [2] Artin, M.: Algebraic approximations of structures over complete local rings. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23-58. · Zbl 0181.48802 [3] Artin, M.: Deformations of singularities. Tata Institute, Bombay, 1976. · Zbl 0395.14003 [4] Atiyah, M. and Macdonald, I. G.: Introduction to commutative algebra. Addison-Wesley, Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0175.03601 [5] Bravo, A., Encinas, S. and Villamayor, O.: A simplified proof of desingularization and applications. Rev. Mat. Iberoamericana 21 (2005), no. 2, 349-458. · Zbl 1086.14012 [6] Bierstone, E. and Milman, P.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128 (1997), no. 7, 207-302. · Zbl 0896.14006 [7] Bierstone, E. and Milman, P.: Desingularization algorithms I. Role of exceptional divisors. Moscow Math. J. 3 (2003), no. 3, 751-805, 1197. · Zbl 1052.14019 [8] Cutkosky, S.: Resolution of singularities. Graduate Studies in Mathematics 63 . Am. Math. Soc., Providence, RI, 2004. · Zbl 1076.14005 [9] Encinas, S. and Hauser, H.: Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77 (2002), no. 4, 821-845. · Zbl 1059.14022 [10] Encinas, S., Nobile, A. and Villamayor, O.: On algorithmic equi-resolution and stratification of Hilbert schemes. Proc. London Math. Soc. 86 (2003), no. 3, 607-648. · Zbl 1076.14020 [11] Encinas, S. and Villamayor, O.: A course on constructive desingularization and equivariance. In Resolution of singularities (Obergurl, 1997) , 147-227. Progr. Math. 181 . Birkhauser, Basel, 2000. · Zbl 0969.14007 [12] Giraud, J.: Sur la théorie du contact maximal. Mat. Z. 137 (1974), 285-310. · Zbl 0275.32003 [13] Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics 52 . Springer-Verlag, New York-Heidelberg, 1977. · Zbl 0367.14001 [14] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero, I-II. Ann. of Math. (2) 79 (1964), 109-326. JSTOR: · Zbl 0122.38603 [15] Hironaka, H.: Idealistic exponent of singularity. In Algebraic geometry (J.J. Sylvester Sympos., John Hopkins Univ., Baltimore, Md., 1976) , 52-125. Johns Hopkins Univ. Press, Baltimore, Md., 1977. · Zbl 0496.14011 [16] Kollar, J.: Resolution of singularities-Seattle lecture. Preprint, [17] Kollar, J.: Lectures on resolution of singularities. Annals of Mathematics Studies 166 . Princeton University Press, Princeton, NJ, 2007. · Zbl 1113.14013 [18] Kunz, E.: Introduction to Commutative Algebra and Algebraic Geometry. Birkhauser Boston, Boston, MA, 1985. · Zbl 0563.13001 [19] Matsumura, H.: Commutative Algebra. W. A. Benjamin, New York, 1970. · Zbl 0211.06501 [20] Matsumura, H.: Commutative ring theory. Cambridge Studies in Advanced Mathematics 8 . Cambridge University Press, Cambridge, 1989. · Zbl 0666.13002 [21] Matsuki, K.: Notes on the inductive algorithm of resolution of singularities of S. Encinas and O. Villamayor. Preprint, [22] Mumford, D.: Lectures on curves on an algebraic surface. Annals of Mathematics Studies 59 . Princeton University Press, Princeton, NJ, 1966. · Zbl 0187.42701 [23] Nobile, A.: A note on flat algebras. Proc. Amer. Math. Soc. 64 (1977), no. 2, 206-208. · Zbl 0407.13006 [24] Schlessinger, M.: Functors of Artin rings. Trans. Amer. Math. Soc. 130 (1968), 208-222. JSTOR: · Zbl 0167.49503 [25] Teissier, B.: Résolution simultanée. In Séminaire sur les sigularités des surfaces. , 71-81. Lecture Notes in Mathematics 777 . Springer, Berlin, 1980. [26] Villamayor, O.: Constructiveness of Hironaka’s resolution. Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1-32. · Zbl 0675.14003 [27] Villamayor, O.: Patching local uniformizations. Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 6, 629-677. · Zbl 0782.14009 [28] Wlodarczyk, J.: Simple Hironaka resolution in characteristic zero. J. Amer. Math. Soc. 18 (2005), no. 4, 779-822. · Zbl 1084.14018 [29] Zariski, O.: Studies in equisingularity, I. Amer. J. Math. 87 (1965), 507-536. JSTOR: · Zbl 0132.41601 [30] Zariski, O.: Studies in equisingularity, II. Amer. J. Math. 87 (1965), 972-1006. JSTOR: · Zbl 0146.42502
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