Comptes Rendus
Algebraic geometry
On morphisms from 3 to 𝔾(1,3)
Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 853-860.

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

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.219
Classification: 14J60, 14M15, 14N05

José Carlos Sierra 1

1 Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, C/Juan del Rosal 10, 28040 Madrid, Spain.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2021__359_7_853_0,
     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},
}
TY  - JOUR
AU  - José Carlos Sierra
TI  - On morphisms from $\protect \mathbb{P}^3$ to $\protect \mathbb{G}(1,3)$
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 853
EP  - 860
VL  - 359
IS  - 7
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.219
LA  - en
ID  - CRMATH_2021__359_7_853_0
ER  - 
%0 Journal Article
%A José Carlos Sierra
%T On morphisms from $\protect \mathbb{P}^3$ to $\protect \mathbb{G}(1,3)$
%J Comptes Rendus. Mathématique
%D 2021
%P 853-860
%V 359
%N 7
%I Académie des sciences, Paris
%R 10.5802/crmath.219
%G en
%F CRMATH_2021__359_7_853_0
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/

[1] Ludovica Chiodera; Philippe Ellia Rank two globally generated vector bundles with c 1 5, Rend. Ist. Mat. Univ. Trieste, Volume 44 (2012), pp. 413-422 | MR | Zbl

[2] Jean D’Almeida Une involution sur un espace de modules de fibrés instantons, Bull. Soc. Math. Fr., Volume 128 (2000) no. 4, pp. 577-584 | DOI | MR | Zbl

[3] Gerhard Edelmann 3-folds in 5 of degree 12, Manuscr. Math., Volume 82 (1994) no. 3-4, pp. 393-406 | DOI | MR | Zbl

[4] David Eisenbud; Joe Harris 3264 and all that. A second course in algebraic geometry, Cambridge University Press, 2016 | DOI | Zbl

[5] Laurent Gruson; Christian Peskine Genre des courbes de l’espace projectif, Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) (Lecture Notes in Mathematics), Volume 687, Springer, 1978, pp. 31-59 | DOI | MR | Zbl

[6] Robin Hartshorne Varieties of small codimension in projective space, Bull. Am. Math. Soc., Volume 80 (1974), pp. 1017-1032 | DOI | MR | Zbl

[7] Robin Hartshorne Stable vector bundles of rank 2 on P 3 , Math. Ann., Volume 238 (1978), pp. 229-280 | DOI | MR | Zbl

[8] Yasuyuki Kachi; Eiichi Sato Segre’s reflexivity and an inductive characterization of hyperquadrics, Memoirs of the American Mathematical Society, 763, American Mathematical Society, 2002 | DOI | Zbl

[9] Shigefumi Mori On degrees and genera of curves on smooth quartic surfaces in 3 , Nagoya Math. J., Volume 96 (1984), pp. 127-132 | DOI | MR | Zbl

[10] Jean-Pierre Serre Sur les modules projectifs, Séminaire Dubreil. Algèbre et théorie des nombres, Volume 14 (1960-1961) no. 1, 2 | Numdam | Zbl

[11] Francesco Severi Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’ suoi punti tripli apparenti, Rend. Circ. Mat. Palermo, Volume 15 (1901), pp. 33-51 | DOI | Zbl

[12] Hiroshi Tango On (n-1)-dimensional projective spaces contained in the Grassmann variety Gr (n,1), J. Math. Kyoto Univ., Volume 14 (1974), pp. 415-460 | DOI | MR | Zbl

[13] Hiroshi Tango On morphisms from projective space P n to the Grassmann variety Gr(n,d), J. Math. Kyoto Univ., Volume 16 (1976), pp. 201-207 | DOI | MR | Zbl

[14] Hiroshi Tango On morphisms from projective space P n to the Grassmann variety Gr(n,d). II, Bull. Kyoto Univ. Educ., Volume 64 (1984), pp. 1-20 | MR | Zbl

Cited by Sources:

Comments - Policy