Projective geometry for blueprints

Javier López Peña; Oliver Lorscheid

doi:10.1016/j.crma.2012.05.001

Algebra/Algebraic Geometry

Projective geometry for blueprints
[Geometrie projective pour les canevas bleus]

Présenté par : Christophe Soulé

Javier López Peña ¹ ; Oliver Lorscheid ²

¹ Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, United Kingdom
² Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germany

Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 455-458.

Résumé
Abstract

Dans cette Note, nous généralisons la Proj-construction des schémas usuels aux schémas bleus. Cela entraine la définition dʼespace projectif et de variétés projectives sur un canevas bleu. En particulier, il est possible de descendre une sous-variété fermée dʼun espace projectif en un $F_{1}$ -modèle canonique. Nous discutons cela dans le cas de la Grassmannienne $Gr (2, 4)$ .

In this Note, we generalize the Proj-construction from usual schemes to blue schemes. This yields the definition of projective space and projective varieties over a blueprint. In particular, it is possible to descend closed subvarieties of a projective space to a canonical $F_{1}$ -model. We discuss this in case of the Grassmannian $Gr (2, 4)$ .

Reçu le : 2012-03-08
Accepté le : 2012-04-27
Publié le : 2012-05-01

DOI : 10.1016/j.crma.2012.05.001

Affiliations des auteurs :

Javier López Peña ¹ ; Oliver Lorscheid ²

¹ Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, United Kingdom
² Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germany

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     title = {Projective geometry for blueprints},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {455--458},
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     doi = {10.1016/j.crma.2012.05.001},
     language = {en},
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Javier López Peña; Oliver Lorscheid. Projective geometry for blueprints. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 455-458. doi : 10.1016/j.crma.2012.05.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.001/

Bibliographie
Cité par

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[2] A. Connes; C. Consani Characteristic 1, entropy and the absolute point, 2009 (preprint) | arXiv

[3] A. Deitmar Schemes over $F_{1}$ , Number Fields and Function Fields—Two Parallel Worlds, Progr. Math., vol. 239, Birkhäuser Boston, Boston, MA, 2005, pp. 87-100

[4] A. Deitmar $F_{1}$ -schemes and toric varieties, Beiträge Algebra Geom., Volume 49 (2008) no. 2, pp. 517-525

[5] K. Kato Toric singularities, Amer. J. Math., Volume 116 (1994) no. 5, pp. 1073-1099

[6] J. López Peña, $F_{1}$ -models for cluster algebras and total positivity, in preparation.

[7] J. López Peña; O. Lorscheid Mapping $F_{1}$ -land an overview of geometries over the field with one element, Noncommutative Geometry, Arithmetic and Related Topics, Johns Hopkins University Press, 2011, pp. 241-265

[8] J. López Peña; O. Lorscheid Torified varieties and their geometries over $F_{1}$ , Math. Z., Volume 267 (2011) no. 3–4, pp. 605-643

[9] O. Lorscheid The geometry of blueprints. Part I: Algebraic background and scheme theory, Adv. Math., Volume 229 (2012) no. 3, pp. 1804-1846

[10] O. Lorscheid The geometry of blueprints. Part II: Tits–Weyl models of algebraic groups, 2012 (preprint) | arXiv

[11] G. Mikhalkin, Tropical geometry, unpublished notes, 2010.

[12] C. Soulé Les variétés sur le corps à un élément, Mosc. Math. J., Volume 4 (2004) no. 1, pp. 217-244 (312)

[13] K. Thas, Notes on $F_{1}$ , I. Combinatorics of $D_{0}$ -schemes and $F_{1}$ -geometry, in preparation.

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