Complex Analytic Desingularization

Hironaka, Heisuke; Vicente, José L.; Aroca, Jose M.; Cordoba, Jose Luis Vicente; Aroca, José Manuel

doi:10.1007/978-4-431-49822-3

Cited by 17 publications

(10 citation statements)

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“…Let the notation be as in 4) above. We have 14) so that π * I is invertible on X. Since we are dealing with a local property, this statement remains valid even if X is not affine.…”

Section: The Universal Mapping Property Of Blowing Upmentioning

confidence: 95%

Handbook of Geometry and Topology of Singularities I

2020

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The problem of resolution of singularities and its solution in various contexts can be traced back to I. Newton and B. Riemann. This paper is an attempt to give a survey of the subject starting with Newton till the modern times, as well as to discuss some of the main open problems that remain to be solved. The main topics covered are the early days of resolution (fields of characteristic zero and dimension up to three), Zariski's approach via valuations, Hironaka's celebrated result in characteristic zero and all dimensions and its subsequent strenthenings and simplifications, existing resutls in positive characteristic (mostly up to dimension three), de Jong's approach via semi-stable reduction, Nash and higher Nash blowing up, as well as reduction of singuarities of vector fields and foliations. In many places, we have tried to summarize the main ideas of proofs of various results without getting too much into technical details.

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“…Let the notation be as in 4) above. We have 14) so that π * I is invertible on X. Since we are dealing with a local property, this statement remains valid even if X is not affine.…”

Section: The Universal Mapping Property Of Blowing Upmentioning

confidence: 95%

Handbook of Geometry and Topology of Singularities I

2020

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“…By a powerful theorem by Hironaka [13] for singular algebraic varieties, the singularities of an arbitrary algebraic variety can always be resolved by a finite number of blow-ups. This is not the case, however, for flows of vector fields, where one can reduce the singularities, but not necessarily complete resolve them (see Chapter 6 in [1]). An example where the sequence of blow-ups does not terminate is given by Smith's equation y ′′ + 4y 3 y ′ + y = 0, which is not of Hamiltonian form.…”

Section: Discussionmentioning

confidence: 99%

“…Apart from the then already known cases of the Lagrange and Euler top, she identified one further integrable case, given by certain ratios of the principle moments of inertia of the top, which is now known as the Kovalevskaya top. P. Painlevé [24] and his pupil B. Gambier [8] took on the challenge of classifying second-order ordinary differential equations of the form (1) y ′′ = R(z, y, y ′ ), R a function rational in y, y ′ with analytic coefficients, with the property now named after Painlevé. The result of this classification was a list of 50 canonical types of equations, in the sense that any equation in the class can be obtained from an equation in the list of 50 by applying a Möbius type transformation…”

Section: Introductionmentioning

confidence: 99%

Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities

Kecker,

Filipuk

2021

Preprint

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In a 1979 paper, K. Okamoto introduced the space of initial values for the six Painlevé equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase space, a rational surface with certain exceptional divisors removed. We show that the construction of the space of initial values remains meaningful for certain classes of second-order complex differential equations, and more generally, Hamiltonian systems, where all movable singularities of all their solutions are algebraic poles (by some authors denoted the quasi-Painlevé property), which is a generalisation of the Painlevé property. The difference here is that the initial value problems obtained in the extended phase space become regular only after an additional change of dependent and independent variables. Constructing the analogue of space of initial values for these equations in this way also serves as an algorithm to single out, from a given class of equations or system of equations, those equations which are free from movable logarithmic branch points.

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“…Proof of Proposition 4.25 (2). We now prove the holonomicity of L by assuming only that G is O S -coherent.…”

Section: Step 2: Reduction After Proper Surjective Generically Finite...mentioning

confidence: 93%