Nesting maps of Grassmannians

De Concini, Corrado; Reichstein, Zinovy

De Concini, Corrado ; Reichstein, Zinovy

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004), p. 109-118 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

Let $F$ be a field and $G r (i, F^{n})$ be the Grassmannian of $i$ -dimensional linear subspaces of $F^{n}$ . A map $f : G r (i, F^{n}) \to G r (j, F^{n})$ is called nesting if $l \subset f (l)$ for every $l \in G r (i, F^{n})$ . Glover, Homer and Stong showed that there are no continuous nesting maps $G r (i, C^{n}) \to G r (j, C^{n})$ except for a few obvious ones. We prove a similar result for algebraic nesting maps $G r (i, F^{n}) \to G r (j, F^{n})$ , where $F$ is an algebraically closed field of arbitrary characteristic. For $i = 1$ this yields a description of the algebraic sub-bundles of the tangent bundle to the projective space $P_{F}^{n}$ .

Sia $F$ un campo e $G r (i, F^{n})$ la Grassmanniana dei sottospazi $i$ -dimensionali di $F^{n}$ . Un’applicazione $f : G r (i, F^{n}) \to G r (j, F^{n})$ si dice «nesting» se $l \subset f (l)$ per ogni $l \in G r (i, F^{n})$ . Glover, Homer and Stong hanno dimostrato che non ci sono applicazioni continue «nesting» da $G r (i, C^{n}) \to G r (j, C^{n})$ a parte un piccolo numero di eccezioni. Dimostriamo un risultato analogo per applicazioni «nesting» algebriche $G r (i, F^{n}) \to G r (j, F^{n})$ , nel caso in cui $F$ sia un campo algebricamente chiuso di caratteristica arbitraria. Per $i = 1$ ciò implica una descrizione dei sottofibrati algebrici del fibrato tangente allo spazio proiettivo $P_{F}^{n}$ .

Publié le : 2004-06-01

MR 2148539 | Zbl 1219.14052

@article{RLIN_2004_9_15_2_109_0,
     author = {Corrado De Concini and Zinovy Reichstein},
     title = {Nesting maps of Grassmannians},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {15},
     year = {2004},
     pages = {109-118},
     zbl = {1219.14052},
     mrnumber = {2148539},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_2004_9_15_2_109_0}
}

De Concini, Corrado; Reichstein, Zinovy. Nesting maps of Grassmannians. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004) pp. 109-118. http://gdmltest.u-ga.fr/item/RLIN_2004_9_15_2_109_0/

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