A Simplified Proof of Desingularization and Applications

Abstract

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

Ideal-Theoretic Strategies for Asymptotic Approximation of Marginal Likelihood Integrals

“…Thus, we can ignore the origin in computing the RLCT. All other points on the exceptional divisor of this blowup lie on some other chart of the blowup where the pullback pair is (s; s 6 ), so the RLCT is at least (7,1). In the chart where…”

“…While general algorithms exist for monomializing any analytic function [7,4], applying them to nonpolynomial functions such as K(ω) can be computationally expensive. In practice, many singular models are parametrized by polynomials.…”

Untitled

“…In our case the order has remained constant at the origin, so one has to consider a finer measure. We will not go into detail but with the help of resolution invariants one can detect an improvement of this singularity; for details see [2,3,15]. Now there is just one chart missing, the z-chart.…”

Today’s menu: Geometry and resolution of singular algebraic surfaces

“…The surfaces live in affine three-space A 3 K = K 3 , where K is a field (of characteristic 0). Mostly we work over the ground field R of real numbers.…”