Planes of matrices of constant rank and globally generated vector bundles

[Plans de matrices de rang constant et fibrés vectoriels globalement engendrés]

Boralevi, Ada ; Mezzetti, Emilia

Annales de l'Institut Fourier, Tome 65 (2015), p. 2069-2089 / Harvested from Numdam

Résumé

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 $c_2\le {{c_1+1}\atopwithdelims ()2}$ est nécessaire et suffisante. Dans le cas $c_1^2-4c_2\ge 0$, 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 $\mathbb{P}^2$ dans une Grassmannienne. Notre résultat étend des travaux antérieurs sur le cas $c_1\le 3$.

Abstract

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 $c_2\le {{c_1+1}\atopwithdelims ()2}$ is necessary and sufficient. For $c_1^2-4c_2\ge 0$, we prove that there exist globally generated bundles, some even defining an embedding of $\mathbb{P}^2$ in a Grassmannian, that cannot correspond to a matrix of the above type. This extends previous work on $c_1\le 3$.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2983
Classification:  14J60,  15A30
Mots clés: Matrices antisymétriques, rang constant, fibrés vectoriels globalement engendrés

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