Planes of matrices of constant rank and globally generated vector bundles
Annales de l'Institut Fourier, Volume 65 (2015) no. 5, p. 2069-2089

We consider the problem of determining all pairs (c 1 ,c 2 ) of Chern classes of rank 2 bundles that are cokernel of a skew-symmetric matrix of linear forms in 3 variables, having constant rank 2c 1 and size 2c 1 +2. We completely solve the problem in the “stable” range, i.e. for pairs with c 1 2 -4c 2 <0, proving that the additional condition c 2 c 1 +1 2 is necessary and sufficient. For c 1 2 -4c 2 0, we prove that there exist globally generated bundles, some even defining an embedding of 2 in a Grassmannian, that cannot correspond to a matrix of the above type. This extends previous work on c 1 3.

On considère le problème de determiner toutes les couples (c 1 ,c 2 ) de classes de Chern de fibrés vectoriels de rang 2 qui sont realisées comme conoyaux de matrices antisymétriques de formes linéaires en trois variables, de taille 2c 1 +2 et rang constant 2c 1 . Le problème est complètement résolu dans le cas “stable”, à savoir lorsque c 1 2 -4c 2 <0, où on démontre que la condition supplémentaire c 2 c 1 +1 2 est nécessaire et suffisante. Dans le cas c 1 2 -4c 2 0, on prouve l’existence de fibrés globalement engendrés qui ne peuvent pas correspondre à des matrices du type ci-dessus, certains même définissant un plongement de 2 dans une Grassmannienne. Notre résultat étend des travaux antérieurs sur le cas c 1 3.

DOI : https://doi.org/10.5802/aif.2983
Classification:  14J60,  15A30
Keywords: Skew-symmetric matrices, constant rank, globally generated vector bundles
@article{AIF_2015__65_5_2069_0,
     author = {Boralevi, Ada and Mezzetti, Emilia},
     title = {Planes of matrices of constant rank and globally generated vector bundles},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {5},
     year = {2015},
     pages = {2069-2089},
     doi = {10.5802/aif.2983},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_5_2069_0}
}
Boralevi, Ada; Mezzetti, Emilia. Planes of matrices of constant rank and globally generated vector bundles. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2069-2089. doi : 10.5802/aif.2983. http://www.numdam.org/item/AIF_2015__65_5_2069_0/

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