The Nash problem for surfaces

Bobadilla, Javier Fernández de; Pereira, María Pe

doi:10.4007/annals.2012.176.3.11

Cited by 39 publications

(57 citation statements)

References 26 publications

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“…Given a variety X , any minimal model program over X , originating from a projective resolution of singularities of X , terminates with a minimal model over X [6]. Minimal models obtained in this way are Q-factorial and are all isomorphic in codimension one.…”

Section: Minimal Modelsmentioning

confidence: 97%

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Terminal valuations and the Nash problem

Fernex

Docampo

2015

Invent. math.

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Let X be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over X. We prove that every terminal valuation over X is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of X. In dimension two, this result gives a new proof of the theorem of Fern\'andez de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.Comment: v2: 21 pages, minor changes and corrections following the referees' reports. To appear in Invent. Mat

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Section: Minimal Modelsmentioning

confidence: 97%

“…Remark 4 6. With the notation as in the proof of Proposition 4.1, it follows by[15, Proposition A.10] that Z • /K ⊗ O Z ∼ = Z /K (and, similarly, Z • /k ⊗ O Z ∼ = Z /k ).…”

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confidence: 92%

Terminal valuations and the Nash problem

Fernex

Docampo

2015

Invent. math.

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“…The Nash problem has a long history. The Nash conjecture turns out to be true for curves, for surfaces [FdBPP12,dFD15], and for several special families of singularities in higher dimensions, including toric varieties [IK03, Ish05, Ish06, GP07, PPP08, LJR12, LA11, LA16]. But there are counterexamples to the Nash conjecture in all dimensions ≥ 3 [IK03,dF13,JK13].…”

Section: Generalities On Arc Spacesmentioning

confidence: 99%

The arc space of the Grassmannian

Docampo¹,

Nigro²

2017

Advances in Mathematics

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Abstract. We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the Grassmannian itself. Just as the combinatorics of Schubert cells is controlled by partitions, the combinatorics in the arc space is controlled by plane partitions (sometimes also called 3d partitions). A combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities. As an application we reduce the computation of log canonical thresholds of pairs involving Schubert varieties to an easy linear programming problem. We also study the Nash problem for Schubert varieties, showing that the Nash map is always bijective in this case.

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“…The most significant recent advance is the proof of Nash's conjecture for surfaces by Fernández de Bobadilla and Pe Pereira [FdBPP12]. Higher dimensional generalizations are proved by de Fernex and Docampo [dFD16].…”

Section: Problem 30mentioning

confidence: 99%

Nash’s work in algebraic geometry

Kollár

2016

Bull. Amer. Math. Soc.

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Abstract. This article is a survey of Nash's contributions to algebraic geometry, focusing on the topology of real algebraic sets and on arc spaces of singularities.Nash wrote two papers in algebraic geometry, one at the beginning of his career [Nash52] and one in 1968, after the onset of his long illness; the latter was published only much later [Nash95]. With these papers Nash was ahead of the times; it took at least 20 years before their importance was recognized. By now both are seen as starting points of significant directions within algebraic geometry. I had the privilege to work in these areas and discuss the problems with Nash. It was impressive that, even after 50 years, these questions were still fresh in his mind and he still had deep insights to share.Section 1 is devoted to the proof of the main theorem of [Nash52], and Section 3 is a short introduction to the study of arc spaces pioneered by [Nash95]. Both of these are quite elementary. Section 2 discusses subsequent work on a conjecture of [Nash52]; some familiarity with algebraic geometry is helpful in reading it. The topology of real algebraic setsOne of the main questions occupying mathematicians around 1950 was to understand the relationships among various notions of manifolds.The Hauptvermutung (=Main Conjecture) formulated by Steinitz and Tietze in 1908 asserted that every topological manifold should be triangulable. The relationship between triangulations and differentiable structures was not yet clear. Whitney proved that every compact C 1 -manifold admits a differentiable-even a real analytic-structure [Whi36], but the groundbreaking examples of Milnor [Mil56,Mil61] were still in the future, and so were the embeddability and uniqueness of real analytic structures [Mor58,Gra58].Nash set out to investigate whether one can find even stronger structures on manifolds. It is quite likely that polynomials form the smallest class of functions that could conceivably be large enough to describe all manifolds. Definition 1. A real algebraic set is the common zero set of a collection of polynomialsComments. For the purposes of this article, one can just think of these as subsets of R N , though theoretically it is almost always better to view X = X(R) as the real points of the complex algebraic set X(C), which consist of all complex solutions of

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The Nash problem for surfaces

Abstract: We prove that Nash mapping is bijective for any surface defined over an algebraically closed field of characteristic 0.

Cited by 39 publications

References 26 publications

Terminal valuations and the Nash problem

Terminal valuations and the Nash problem

The arc space of the Grassmannian

Nash’s work in algebraic geometry

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