This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i. e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme X is achieved by using the procedure of principalization for the ideal I(X) associated to the embedded scheme X. The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.Part 3. Applications. 21 7. Weak and strict transforms of ideals: Strong Factorizing Desingularization. 21 8. On a class of regular schemes and on real and complex analytic spaces. 31 9. Non-embedded desingularization. 34 10. Equiresolution. Families of schemes. 37 11. Bodnár-Schicho's computer implementation. 43Example 3.3. If X ⊂ W is a hypersurface and J is its defining ideal, then Sing(J, b) is the set of points of X where the multiplicity is greater than or equal to b.Example 3.4. If J ⊂ O W is an arbitrary non-zero sheaf of ideals, then Sing(J, b) is the set of points of W where the order of J is greater than or equal to b.for the pair (W, E) and Y ⊂ Sing(J, b). 3.6. Permissible transformations of basic objects. Let (W, (J, b), E = {H 1 , . . . , H r }) be a basic object and let Y ⊂ Sing(J, b) be a permissible center. Consider W ←− W 1 the monoidal transformation with center Y . This induces a transformation of pairs,(3.6.1)as the permissible transformation of the basic object (W, (J, b), E).Remark 3.7. In general, given a sequence of transformations of basic objects (3.7.1)at centers Y i ⊂ Sing(J i , b), i = 0, 1, . . . k − 1, we obtain expressionsNote here that c r+1 = . . . = c r+i = b if none of the centers Y i are included in any of the exceptional divisors H j .Definition 3.8. A finite sequence transformation of basic objects as (3.7.1) is a resolution of (W 0 ,Remark 3.9. Note that:(1) If sequence (3.7.1) is a resolution of the basic object (W 0 ,where M k is an invertible sheaf of ideals, and J k has no points of order ≥ b in W k . (2) The ideal J k is not the strict transform of J 0 , an ideal which is far more complicated to define (see Section 7 for a discussion on this matter). However it is so in some particular cases. In fact, if X 0 ⊂ W 0 is a closed smooth subscheme, and J 0 = I(X), then Sing(J 0 , 1) = X, and given any sequence of transformations of basic objects
Patching local uniformizationsAnnales scientifiques de l'É.N.S. 4 e série, tome 25, n o 6 (1992), p. 629-677
We present results on multiplicity theory. Differential operators on smooth schemes play a central role in the study of the multiplicity of an embedded hypersurface at a point. This follows from the fact that the multiplicity is defined by the Taylor development of the defining equation at such point; and the Taylor development involves higher order differentials. On the other hand, the multiplicity of a hypersurface can be expressed in terms of general projections defined at étale neighborhoods of such point: Given a local embedding of a (d − 1)-dimensional hypersurface in a smooth d-dimensional scheme, a general projection on a smooth (d − 1)-dimensional scheme is a finite map on the hypersurface; and the multiplicity at the point is the degree of this finite map.We relate both approaches. In fact we study invariants of embedded hypersurfaces, defined in terms of differential operators, which express properties of the ramification of the morphism. Of particular interest is the case of hypersurfaces over fields of positive characteristic.A central result in multiplicity theory of hypersurfaces is a form of elimination of one variable in the description of highest multiplicity locus. This form of elimination, known over fields of characteristic zero, is achieved with the notion of Tschirnhausen polynomial introduced by Abhyankar. This is a key point in the proof of embedded desingularization.In this paper we provide a characteristic free approach to this form of elimination. Our alternative approach is based on projections on smooth (d − 1)-dimensional schemes.The properties of this new form of elimination remain weaker in positive characteristic, than it does in characteristic zero, when it comes to compatibility of elimination with permissible monoidal transformation. This opens the way to new questions on resolution problems. We also discuss the behavior of invariants, attached to a singularity at a point, with this form of elimination.
We present a proof of embedded desingularization for closed subschemes which does not make use of Hilbert-Samuel function and avoids Hironaka's notion of normal flatness (see also [17] page 224). Given a subscheme defined by equations, we prove that embedded desingularization can be achieved by a sequence of monoidal transformations; where the law of transformation on the equations defining the subscheme is simpler then that used in Hironaka's procedure. This is done by showing that desingularization of a closed subscheme X, in a smooth sheme W, is achieved by taking an algorithmic principalization for the ideal I(X), associated to the embedded scheme X. This provides a conceptual simplification of the original proof of Hironaka. This algorithm of principalization (of Logresolution of ideals), and this new procedure of embedded desingularization discussed here, have been implemented in MAPLE.
Let V be a smooth scheme over a field k, and let {I n , n ≥ 0} be a filtration of sheaves of ideals in O V , such that I 0 = O V , and I s · I t ⊂ I s+t . In such case I n is called a Rees algebra. A Rees algebra is said to be a differential algebra if, for any two integers N > n and any differential operator D of order n, D(I N ) ⊂ I N −n . Any Rees algebra extends to a smallest differential algebra.There are two extensions of Rees algebras of interest in singularity theory: one defined by taking integral closures, and another by extending the algebra to a differential algebra.We study here some relations between these two extensions, with particular emphasis on the behavior of higher order differentials over arbitrary fields.
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