Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents
[Problème d'équivalence pour des courbes rationnelles minimales à variétés des tangentes rationnelles minimales isotriviales]
Hwang, Jun-Muk
Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010), p. 607-620 / Harvested from Numdam
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Nous énonçons le problème d’équivalence, au sens de É. Cartan, pour des familles de courbes rationnelles minimales sur des variétés projectives uniréglées. Un invariant important de ce problème d’équivalence est la variété des tangentes rationnelles minimales. Nous étudions le cas où les variétés de tangentes rationnelles minimales aux points génériques forment une famille isotriviale. La question principale dans ce cas est  : pour quelle variété projective Z une famille de courbes rationnelles minimales, dont les variétés de tangentes rationnelles minimales sont Z-isotriviales, est-elle localement équivalente au modèle plat  ? Nous montrons que c’est le cas lorsque Z vérifie certaines conditions de géométrie projective qui sont satisfaites pour une hypersurface non singulière de degré ≥4.

We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety Z, a family of minimal rational curves with Z-isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when Z satisfies certain projective-geometric conditions, which hold for a non-singular hypersurface of degree ≥4.

Publié le : 2010-01-01
DOI : https://doi.org/10.24033/asens.2129
Classification:  58A15,  14J40,  53B99
Mots clés: problème d'équivalence, courbes rationnelles minimales 
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     author = {Hwang, Jun-Muk},
     title = {Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {43},
     year = {2010},
     pages = {607-620},
     doi = {10.24033/asens.2129},
     mrnumber = {2722510},
     zbl = {1210.14044},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2010_4_43_4_607_0}
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Hwang, Jun-Muk. Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents. Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010) pp. 607-620. doi : 10.24033/asens.2129. http://gdmltest.u-ga.fr/item/ASENS_2010_4_43_4_607_0/

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