A Simplified Proof of Desingularization and Applications (original) (raw)

Desingularization algorithms I. Role of exceptional divisors

Moscow Mathematical Journal

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.

A strong desingularization theorem

Arxiv preprint math/0104001, 2001

Abstract: Let $ X $ be a closed subscheme embedded in a scheme $ W $ smooth over a field bfk{\ bf k} bfk of characteristic zero, and let mathcalI(X){\ mathcal I}(X) mathcalI(X) be the sheaf of ideals defining $ X .Assumethatthesetofregularpointsof. Assume that the set of regular points of .Assumethatthesetofregularpointsof X $ is dense in $ X $. We prove that there ...

Good points and constructive resolution of singluarities

Acta Mathematica, 1998

To the memory of Professor Manfred Herrmann X (P3) For any (X, W), if Im~={al, ..., a,} then =Ui=x ~) 1(Oli) is a stratification of X, each stratum f-l(ai) being locally closed, regular and of pure dimension.

Theory of Schemes II

These are notes from Mathematics 233br, an advanced graduate seminar on schemes, taught by Dr. Junecue Suh during the Spring of 2014 at Harvard University. Please excuse the roughness and brevity of some of the sections. Any errors found in these notes should be attributed to the scribe. The first half of the course was concerned with derived categories, primarily following the text of Gelfand and Manin ([2]). Since our lectures did not differ in any significant way from their treatment, I did not deem it necessary to write up those notes. Before proceeding, the reader should be acquainted with the content of this text, the first chapters of which are crucial to understanding the following lectures.

Part 1. Duality and Flat Base Change on Formal Schemes

Contemporary Mathematics, 1999

In §8.3 of our paper "Duality and Flat Base Change on Formal Schemes" [DFS] some important results concerning localization of, and preservation of coherence by, basic duality functors, were based on the false statement that any closed formal subscheme of an open subscheme of the completion P of a relative projective space is an open subscheme of a closed formal subscheme of P. In this note, the said results are provided with solid foundations. In Proposition 8.3.1 of our paper [DFS], the duality functors f ! and f # associated to a pseudo-proper map f : X → Y of noetherian formal schemes (i.e., right adjoints of suitable restrictions of the derived direct-image functor Rf *) are asserted to be local on X, as a consequence of flat base change. Moreover, in Proposition 8.3.2 it is asserted that (roughly speaking) f # preserves coherence. Brian Conrad pointed out that our justifications are deficient because they use the claim 8.3.1(c) that a map between noetherian formal schemes that can be factored as a closed immersion followed by an open one can also be factored as an open immersion followed by a closed one, which is not true in general. 1 Indeed, Conrad observed that for any (A, x, p) with A an adic domain, x ∈ A such that B := A {x} is a domain, and p a nonzero B-ideal contracting to (0) in A, the natural map Spf(B/p) → Spf(A) is a counterexample. Such a triple was provided to us by Bill Heinzer: With w, x, y, z indeterminates over a field k, set A := k[w, x, z][[y]] and B := A {x} = k[w, x, 1/x, z][[y]]. Let P be the prime ideal (w, z)A and R := A P ⊂ B P B = : S, so that R ⊂ S are 2-dimensional regular local domains such that the residue field of S (i.e., the fraction field of k[x, 1/x][[y]]) is transcendental over that of R (i.e., the fraction field of k[x][[y]]). Then [HR, p. 364, Theorem 1.12] says that there exist infinitely many height-one prime S-ideals in the generic fiber over R. Any of these contracts in B to a (prime) p as above.

A Strengthening of Resolution of Singularities in Characteristic Zero

Proceedings of the London Mathematical Society, 2003

Let XXX be a closed subscheme embedded in a scheme WWW, smooth over a field bfk{\bf k}bfk of characteristic zero, and let mathcalI(X){\mathcal I} (X)mathcalI(X) be the sheaf of ideals defining XXX. Assume that the set of regular points of XXX is dense in XXX. We prove that there exists a proper, birational morphism, pi:WrlongrightarrowW\pi : W_r \longrightarrow Wpi:WrlongrightarrowW, obtained as a composition of monoidal transformations, so that if XrsubsetWrX_r \subset W_rXrsubsetWr denotes the strict transform of XsubsetWX \subset WXsubsetW then:(1) the morphism pi:WrlongrightarrowW\pi : W_r \longrightarrow Wpi:WrlongrightarrowW is an embedded desingularization of XXX (as in Hironaka's Theorem);(2) the total transform of mathcalI(X){\mathcal I} (X)mathcalI(X) in mathcalOWr{\mathcal O}_{W_r}mathcalOWr factors as a product of an invertible sheaf of ideals mathcalL{\mathcal L}mathcalL supported on the exceptional locus, and the sheaf of ideals defining the strict transform of XXX (that is, mathcalI(X)mathcalOWr=mathcalLcdotmathcalI(Xr){\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)mathcalI(X)mathcalOWr=mathcalLcdotmathcalI(Xr)).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingulariza...