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

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[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

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

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