Desingularization of Two-Dimensional Schemes

Lipman, Joseph

doi:10.2307/1971141

Cited by 303 publications

(191 citation statements)

References 22 publications

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“…Proposition 8 There exists a minimal nonsingular model Proof: Since any algebraic k-surface admits a minimal resolution of singularities (A. p.155, [Li2]), X can be replaced with its minimal resolution X ′ and Y by the minimal resolution Y ′ of the normalization of X in L. It hence can be assumed that f : Y → X is nonsingular. Let f ′ : Y ′ → X ′ be a nonsingular model dominating f .…”

Section: The Algorithmmentioning

confidence: 99%

Monomial resolutions of morphisms of algebraic surfaces

Cutkosky

Piltant

2000

Communications in Algebra

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Dedicated to Robin Hartshorne on the occasion of his sixtieth birthday 1 Introduction.Let k be a perfect field and L/K be a finite separable field extension of onedimensional function fields over k. A classical result (c.f. I.6, [Ha]) states that K (resp. L) has a unique proper and smooth model C (resp. D), and that there is a unique morphism of curves f : D → C inducing the field inclusion K ⊂ L at the generic points of C and D. It has the following properties:(i) f is a finite morphism.(ii) f is monomial on its tamely ramified locus; let β ∈ D be any point, with α := f (β) ∈ C, such that the extension of discrete valuation rings O Y,β /O X,α is tamely ramified. There exists a local-étale ring extension R of O Y,β and regular parameters u of O X,α and x of R such that u = x a for some a prime to the characteristic of k.

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Section: The Algorithmmentioning

confidence: 99%

Monomial resolutions of morphisms of algebraic surfaces

Cutkosky

Piltant

2000

Communications in Algebra

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“…-We can find a finite morphism S" ->• S of an integral scheme 6" to S', and a finite alteration X' -> X with purely inseparable function field extension such that X' is a scheme over 5" with geometrically irreducible geometric fibre. Since dim 5" ^ 2, we have canonical resolution of singularities for S' [4] and any alteration of S", and hence we get (5.12.1) for S". By the theorem we get (5.12.1) for X'.…”

Section: -Let F : X -> S Be a Dominant And Proper Morphism Of Integramentioning

confidence: 85%

Families of curves and alterations

Jong¹

1997

Annales De L’institut Fourier

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“…There are published proofs for the case of qe surfaces, see e.g. [Lip78], and for the case of k-varieties of dimension at most 3 under the minor assumption that the imperfection rank of k is finite (i.e., if p = char(k) > 0 then [k : k p ] < ∞), see [CP09]. In the preprint [CP14], Cossart and Piltant extend their method from [CP09] to arbitrary qe threefolds, and experts in the desingularization theory think that the proof is correct.…”

Section: Main Theoremmentioning

confidence: 99%

Tame distillation and desingularization by $p$-alterations

Temkin

2017

Ann. of Math. (2)

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Abstract. We strengthen Gabber's l ′ -alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme X of finite type over a quasi-excellent threefold can be desingularized by a char(X)-alteration, i.e. an alteration whose order is only divisible by primes noninvertible on X. The main new ingredient in the proof is a tame distillation theorem asserting that, after enlarging, any alteration of X can be split into a composition of a tame Galois alteration and a char(X)-alteration. The proof of the distillation theorem is based on the following tameness theorem that we deduce from a theorem of M. Pank: if a valued field k of residue characteristic p has no non-trivial p-extensions then any algebraic extension l/k is tame.1. Introduction 1.1. Background. This paper falls within the area of resolution of singularities by alterations, so we start with a brief review of known altered desingularization results.1.1.1. de Jong's theorems. In [dJ96, Theorem 4.1] Johan de Jong proved that, regardless of the characteristic of the ground field, an integral variety X can be desingularized by an alteration b : X ′ → X, i.e. a proper dominant generically finite morphism between integral schemes. In addition, given a closed subset Z X one can achieve that Z ′ = b −1 (Z) is a simple normal crossings (snc) divisor, and f can be chosen G-Galois in the sense that the alteration X ′ /G → X is generically radicial, where G = Aut X (X ′ ), see [dJ96, Theorem 7.3]. de Jong's altered desingularization was the first resolution result that applies in such generality and it immediately found numerous applications. Also, de Jong proved an altered version of semistable reduction for an integral scheme X over an excellent curve S, see [dJ96, Theorem 8.2]. The latter can be viewed as altered desingularization of morphisms f : X → S with S a curve. IT14b]. The most recent and powerful advance is Gabber's l ′ -altered desingularization: it is proved in [IT14b, Theorems 2.1 and 2.4] that if l is a prime invertible on X then one can achieve that the degree [k(X ′ ) : k(X)] of the alteration b is not divisible by l both in the altered desingularization and altered semistable reduction theorems. This extended the field of applications of altered desingularization to cohomology theories with coefficients where l is not inverted, for example, Z/lZ or Z l .

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Desingularization of Two-Dimensional Schemes

Cited by 303 publications

References 22 publications

Monomial resolutions of morphisms of algebraic surfaces

Monomial resolutions of morphisms of algebraic surfaces

Families of curves and alterations

Tame distillation and desingularization by $p$-alterations

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