arXiv subjects
Publications and source records attributed to Edward Bierstone.
Building upon works of Hironaka, Bierstone-Milman, Villamayor and Wlodarczyk, we give an a priori estimate for the complexity of the simplified Hironaka algorithm. As a consequence of this result, we show that there exists canonical Hironaka embedded desingularization and principalization over fields of large characteristic (relative to the degrees of generating polynomials).
Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X,D) is said to be semi-simple normal crossings (semi-snc) at a point a of X if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up f: X' -> X such that the transformed pair (X',D') is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X,D). The result answers a question of Kollar.
The philosophy of the article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the smallest class of singularities (in low dimension or low codimension) with which we necessarily end up if we avoid blowing up normal crossings singularities. Several of the questions studied were raised by Kollar.
In this sequel to Resolution except for minimal singularities I, we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities.
We present a constructive criterion for flatness of a morphism of analytic spaces X -> Y or, more generally, for flatness over Y of a coherent sheaf of modules on X. The criterion is a combination of a simple linear-algebra condition "in codimension zero" and a condition "in codimension one" which can be used together with the Weierstrass preparation theorem to inductively reduce the fibre dimension of the morphism.
We prove that the algorithm for desingularization of algebraic varieties in characteristic zero of the first two authors is functorial with respect to regular morphisms. For this purpose, we show that, in characteristic zero, a regular morphism with connected affine source can be factored into a smooth morphism, a ground-field extension and a generic-fibre embedding. Every variety of characteristic zero admits a regular morphism to a Q-variety. The desingularization algorithm is therefore Q-universal or absolute in the sense that it is induced from its restriction to varieties over Q. As a consequence, for example, the algorithm extends functorially to localizations and Henselizations of varieties.
Algorithms for resolution of singularities in characteristic zero are based on Hironaka's idea of reducing the problem to a simpler question of desingularization of an "idealistic exponent" (or "marked ideal"). How can we determine whether two marked ideals are equisingular in the sense that they can be resolved by the same blowing-up sequences? We show there is a desingularization functor defined on the category of equivalence classes of marked ideals and smooth morphisms, where marked ideals are "equivalent" if they have the same sequences of "test transformations". Functoriality in this sense realizes Hironaka's idealistic exponent philosophy. We use it to show that the recent algorithms for desingularization of marked ideals of Wlodarczyk and of Kollar coincide with our own, and we discuss open problems. This article is dedicated to Heisuke Hironaka for his 77th birthday, in celebration of "Kiju" -- joy and long life!
We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that satisfy the normal flatness criterion of Hironaka. The results extend to more general varieties defined locally by binomial equations.
The article is about a "desingularization principle" common to various canonical desingularization algorithms in characteristic zero, and the roles played by the exceptional divisors in the underlying local construction. We compare algorithms of the authors and of Villamayor and his collaborators, distinguishing between the fundamental effect of the way the exceptional divisors are used, and different theorems obtained because of flexibility allowed in the choice of "input data". We show how the meaning of "invariance" of the desingularization invariant, and the efficiency of the algorithm depend on the notion of "equivalence" of collections of local data used in the inductive construction.
Let g denote a real analytic function on an open subset U of Euclidean space, and let S denote the boundary points of U where g does not admit a local analytic extension. We show that if g is semialgebraic (respectively, globally subanalytic), then S is semialgebraic (respectively, subanalytic) and g extends to a neighbourhood of cl(U)§as an analytic function that is semialgebraic (respectively, globally subanalytic). (In the general subanalytic case, S is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series G centred at points in the image of an analytic mapping, in terms of the radii of convergence of the pull-backs of G at points of the source.
We show that a version of the desingularization theorem of Hironaka holds for certain classes of infinitely differentiable functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the Denjoy-Carleman theorem. These classes have been poorly understood in dimension > 1. Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, Lojasiewicz inequalities, division properties.
This article is an exposition of an elementary constructive proof of canonical resolution of singularities in characteristic zero, presented in detail in Invent. Math. 128 (1997), 207-302. We define a new local invariant and get an algorithm for canonical desingularization by successively blowing up its maximum loci. The invariant can be described by local computations that provide equations for the centres of blowing up. We describe the origin of our approach and present the proof (in the hypersurface case) in parallel with a worked example.
This article contains an elementary constructive proof of resolution of singularities in characteristic zero. Our proof applies in particular to schemes of finite type and to analytic spaces (so we recover the great theorems of Hironaka). We introduce a discrete local invariant $\inv_X (a)$ whose maximum locus determines a smooth centre of blowing up, leading to desingularization. To define $\inv_X$, we need only to work with a category of local-ringed spaces $X=(|X|,{\cal O}_X)$ satisfying certain natural conditions. If $a\in |X|$, then $\inv_X(a)$ depends only on $\widehat{\cal O}_{X,a}$. More generally, $\inv_X(a)$ is defined inductively after any sequence of blowings-up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of $\inv_X(\cdot)$. The paper is self-contained and includes detailed examples. One of our goals is that the reader understand the desingularization theorem, rather than simply ``know'' it is true.
We introduce a new point of view towards Glaeser's theorem on composite $C^\infty$ functions [Ann. of Math. 1963], with respect to which we can formulate a ``$C^k$ composite function property" that is satisfied by all semiproper real analytic mappings. As a consequence, we see that a closed subanalytic set $X$ satisfies the $C^\infty$ composite function property if and only if the ring $C^\infty (X)$ of $C^\infty$ functions on $X$ is the intersection of all finite differentiability classes.