On morphisms from $\protect \mathbb{P}^3$ to $\protect \mathbb{G}(1,3)$

José Carlos Sierra

doi:10.5802/crmath.219

Géométrie algébrique

On morphisms from

ℙ^{3}

to

𝔾 (1, 3)

José Carlos Sierra ¹

¹ Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, C/Juan del Rosal 10, 28040 Madrid, Spain.

Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 853-860.

Résumé

Every morphism from $ℙ^{n}$ to $𝔾 (k, m)$ is constant if $m < n$ , and nonconstant morphisms from $ℙ^{n}$ to $𝔾 (k, n)$ rarely appear when $0 < k < n - 1$ . In this setting, Tango proved that a morphism from $ℙ^{n}$ to $𝔾 (1, n)$ is constant if $n \notin {3, 5}$ . Here we focus on the case $n = 3$ and show that if $ϕ : 𝒪_{ℙ^{3}}^{\oplus 4} \to E$ is the surjection onto a rank $2$ vector bundle $E$ inducing a morphism $φ : ℙ^{3} \to 𝔾 (1, 3)$ , then $h^{1} (E^{*}) \leq 1$ . Furthermore, a complete classification is given if equality holds.

Reçu le : 2021-02-12
Accepté le : 2021-04-29
Publié le : 2021-09-17

DOI : 10.5802/crmath.219

Classification : 14J60, 14M15, 14N05

Affiliations des auteurs :

José Carlos Sierra ¹

¹ Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, C/Juan del Rosal 10, 28040 Madrid, Spain.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Jos\'e Carlos Sierra},
     title = {On morphisms from $\protect \mathbb{P}^3$ to $\protect \mathbb{G}(1,3)$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {853--860},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {7},
     year = {2021},
     doi = {10.5802/crmath.219},
     language = {en},
}

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José Carlos Sierra. On morphisms from $\protect \mathbb{P}^3$ to $\protect \mathbb{G}(1,3)$. Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 853-860. doi : 10.5802/crmath.219. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.219/

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