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},
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     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.

[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

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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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Manoel Jarra; Oliver Lorscheid Flag matroids with coefficients, Advances in Mathematics, Volume 436 (2024), p. 109396 | DOI:10.1016/j.aim.2023.109396
Matthew Baker; Oliver Lorscheid The moduli space of matroids, Advances in Mathematics, Volume 390 (2021), p. 107883 | DOI:10.1016/j.aim.2021.107883
Koen Thas Projective spaces over, Journal of Combinatorial Designs, Volume 27 (2019) no. 2, p. 55 | DOI:10.1002/jcd.21639
Oliver Lorscheid F 1 $F_{1}$ for Everyone, Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 120 (2018) no. 2, p. 83 | DOI:10.1365/s13291-018-0177-x
Oliver Lorscheid; Matt Szczesny Quasicoherent sheaves on projective schemes overF1, Journal of Pure and Applied Algebra, Volume 222 (2018) no. 6, p. 1337 | DOI:10.1016/j.jpaa.2017.07.001
Oliver Lorscheid Blue schemes, semiring schemes, and relative schemes after Toën and Vaquié, Journal of Algebra, Volume 482 (2017), p. 264 | DOI:10.1016/j.jalgebra.2017.03.023
Manuel Mérida-Angulo; Koen Thas Deitmar schemes, graphs and zeta functions, Journal of Geometry and Physics, Volume 117 (2017), p. 234 | DOI:10.1016/j.geomphys.2017.01.027

Cité par 7 documents. Sources : Crossref

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