Automated Resolution of Singularities for Hypersurfaces

Bodnár, Gábor; Schicho, Josef

doi:10.1006/jsco.1999.0414

Cited by 27 publications

(34 citation statements)

References 17 publications

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“…You are invited to sharpen your pencil. More examples can be found in [EV 2], [BM3], [BM5], [BS1], [BS 3]. As the number of charts increases quickly after each blowup, we shall restrict ourselves sometimes to the most interesting points of the exceptional divisor and thus compute only local resolutions (more precisely, the local data of the resolution along a certain valuation).…”

Section: Examplesmentioning

confidence: 99%

The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

Hauser

2003

Bull. Amer. Math. Soc.

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Abstract. This paper is a handyman's manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups.Three objectives: Pleasant writing, easy reading, good understanding.One topic: How to prove resolution of singularities in characteristic zero. Statement to be proven (No-Tech):The solutions of a system of polynomial equations can be parametrized by the points of a manifold. Statement to be proven (Low-Tech):The zero-set X of finitely many real or complex polynomials in n variables admits a resolution of its singularities (we understand by singularities the points where X fails to be smooth). The resolution is a surjective differentiable map ε from a manifoldX to X which is almost everywhere a diffeomorphism, and which has in addition some nice properties (e.g., it is a composition of especially simple maps which can be explicitly constructed). Said differently, ε parametrizes the zero-set X (see Figure 1). Figure 1. Singular surface Ding-dong:The zero-set of the equation x 2 + y 2 = (1 − z)z 2 in R 3 can be parametrized by R 2 via (s, t) → (s(1 − s 2 ) · cos t, s(1 − s 2 ) · sin t, 1 − s 2 ). The picture shows the intersection of the Ding-dong with a ball of radius 3. You will agree that such a parametrization is particularly useful, either to produce pictures of X (at least in small dimensions), or to determine geometric and topological properties of X. The huge number of places where resolutions are applied to prove theorems about all types of objects (algebraic varieties, compactifications, diophantine equations, cohomology groups, foliations, separatrices, differential equations, D-modules, distributions, dynamical systems, etc.) shows that the existence of resolutions is really basic to many questions. But it is by no means an easy matter to construct a resolution for a given X. Puzzle:Here is an elementary problem in combinatorics -the polyhedral game of Hironaka. Finding a winning startegy for it is instrumental for the way singularities will be resolved. Each solution to the game can yield a different method of resolution. The formulation is simple.Given are a finite set of points A in N n , with positive convex hull N in R n (see Figure 2),(0,0) There are two players, P 1 and P 2 . They compete in the following game. Player P 1 starts by choosing a non-empty subset J of {1, . . . , n}. Player P 2 then picks a number j in J.After these "moves", the set A is replaced by the set A obtained from A by substituting the j-th component of vectors α in A by the sum of the components α i with index i in J, say α j → α j = i∈J α i . The other components remain untouched, α k = α k for k = j (see Figure 3). Then set N = conv(A ) + R n ≥0 . The next round starts over again, with N replaced by N : P 1 chooses a subset J of {1, . . . , n}, and player P 2 picks j in J as before. The polyhedron N is replaced by the corresponding polyhedron N . In this way, the game continues. THE HIRONAKA THEOREM ON RESOLUTION OF SINGULARITIES 325Player P ...

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Section: Examplesmentioning

confidence: 99%

The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

Hauser

2003

Bull. Amer. Math. Soc.

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“…We refer the reader to [8] We will refer to it as the resolution defined by the algorithm, or the resolution defined by the functions f i . Note that B says that this resolution is equivariant.…”

Section: 5mentioning

confidence: 99%

A new Proof of Desingularization over fields of characteristic zero

Encinas¹,

Villamayor²

2003

Rev. Mat. Iberoam.

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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.

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“…Now we describe two fundamental operations on varieties which are, for instance, also elementary steps of the resolution algorithm in Bodnár and Schicho (2000a). The Cover operation replaces the input chart U by the complements of the zero sets of f 1 , .…”

Section: Elimination Of Chart Redundanciesmentioning

confidence: 99%

Two Computational Techniques for Singularity Resolution

Bodnár

Schicho²

2001

Journal of Symbolic Computation

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This paper presents two efficient computational techniques in algebraic geometry. The first one allows the elimination of redundancies in the representation of quasi-projective varieties by atlases of affine charts. The second simplifies the computations with exponentiated ideals by attaching rational weights to the generators, applying Hironaka's theory of idealistic exponents. As the main application, we used these techniques to speed up Villamayor's algorithm for resolving hypersurface singularities in any dimension.

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Automated Resolution of Singularities for Hypersurfaces

Abstract: We give the description of an algorithm for the resolution of singularities, in the case of hypersurfaces in characteristic zero, based on Villamayor's stratifying function. The algorithm is implemented in Maple.

Cited by 27 publications

References 17 publications

The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

A new Proof of Desingularization over fields of characteristic zero

Two Computational Techniques for Singularity Resolution

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