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H 2 Arc spaces and singularities in the minimal model program - Lecture 1

Auteurs : de Fernex, Tommaso (Auteur de la Conférence)
CIRM (Editeur )

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arc space of a variety arc valuation arc spaces - examples and remarks irreducibility of the arc space Kolchin theorem Nash theorem Nash valuation Nash map Nash problem Questions

Résumé : The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture.

Codes MSC :
13A18 - Valuations and their generalizations [See also 12J20]
14B05 - Singularities
14E15 - Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]
14E30 - Minimal model program (Mori theory, extremal rays)
14E18 - Arcs and motivic integration

    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 26/02/15
    Date de captation : 09/02/15
    Collection : Research talks
    Format : QuickTime (.mov) Durée : 01:02:02
    Domaine : Algebraic & Complex Geometry
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2015-02-09_Fernex_part1.mp4

Informations sur la rencontre

Nom du congrès : Young researchers in singularities / Jeunes chercheurs en singularités
Organisteurs Congrès : Fichou, Goulwen ; Plénat, Camille
Dates : 09/02/15 - 13/02/15
Année de la rencontre : 2015
URL Congrès : http://chairejeanmorlet-1stsemester2015....

Citation Data

DOI : 10.24350/CIRM.V.18697103
Cite this video as: de Fernex, Tommaso (2015). Arc spaces and singularities in the minimal model program - Lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18697103
URI : http://dx.doi.org/10.24350/CIRM.V.18697103

Voir aussi


  1. [1] de Fernex, T. (2013). Three-dimensional counter-examples to the Nash problem. Compositio Mathematica, 149(9), 1519-1534 - http://dx.doi.org/10.1112/S0010437X13007252

  2. [2] de Fernex, T., & and Docampo, R. (2014). Terminal valuations and the Nash problem. - http://arxiv.org/abs/1404.0762

  3. [3] Ein, L., Lazarsfeld, R., & Mustata, M. (2004). Contact loci in arc spaces. Compositio Mathematica, 140(5), 1229-1244 - http://dx.doi.org/10.1112/S0010437X04000429

  4. [4] Ein, L., Mustata, M., & Yasuda, T. (2003). Jet schemes, log discrepancies and inversion of adjunction. Inventiones Mathematicae, 153(3), 519-535 - http://dx.doi.org/10.1007/s00222-003-0298-3

  5. [5] Fernández de Bobadilla, J., & Pe Pereira, M. (2012). The Nash problem for surfaces. Annals of Mathematics. Second Series, 176(3), 2003-2029 - http://dx.doi.org/10.4007/annals.2012.176.3.11

  6. [6] Ishii, S., & Kollár, J. (2003). The Nash problem on arc families of singularities. Duke Mathematical Journal, 120(3), 601-620 - http://dx.doi.org/10.1215/S0012-7094-03-12034-7

  7. [7] Nash, John F. jun. (1995). Arc structure of singularities. Duke Mathematical Journal, 81(1), 31-38 - http://dx.doi.org/10.1215/S0012-7094-95-08103-4

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