Planes of matrices of constant rank and globally generated vector bundles

Boralevi, Ada; Mezzetti, Emilia

doi:10.5802/aif.2983

Planes of matrices of constant rank and globally generated vector bundles
[Plans de matrices de rang constant et fibrés vectoriels globalement engendrés]

Boralevi, Ada ¹ ; Mezzetti, Emilia ²

¹ Scuola Internazionale Superiore di Studi Avanzati via Bonomea 265 34136 Trieste (Italy)
² Dipartimento di Matematica e Geoscienze Sezione di Matematica e Informatica Università di Trieste Via Valerio 12/1 34127 Trieste (Italy)

Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2069-2089.

Résumé
Abstract

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 $2 c_{1} + 2$ et rang constant $2 c_{1}$ . Le problème est complètement résolu dans le cas “stable”, à savoir lorsque $c_{1}^{2} - 4 c_{2} < 0$ , où on démontre que la condition supplémentaire $c_{2} \leq (\binom{c_{1} + 1}{2})$ est nécessaire et suffisante. Dans le cas $c_{1}^{2} - 4 c_{2} \geq 0$ , 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 $ℙ^{2}$ dans une Grassmannienne. Notre résultat étend des travaux antérieurs sur le cas $c_{1} \leq 3$ .

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 $2 c_{1}$ and size $2 c_{1} + 2$ . We completely solve the problem in the “stable” range, i.e. for pairs with $c_{1}^{2} - 4 c_{2} < 0$ , proving that the additional condition $c_{2} \leq (\binom{c_{1} + 1}{2})$ is necessary and sufficient. For $c_{1}^{2} - 4 c_{2} \geq 0$ , we prove that there exist globally generated bundles, some even defining an embedding of $ℙ^{2}$ in a Grassmannian, that cannot correspond to a matrix of the above type. This extends previous work on $c_{1} \leq 3$ .

DOI : 10.5802/aif.2983

Classification : 14J60, 15A30
Keywords: Skew-symmetric matrices, constant rank, globally generated vector bundles
Mot clés : Matrices antisymétriques, rang constant, fibrés vectoriels globalement engendrés

Affiliations des auteurs :

Boralevi, Ada ¹ ; Mezzetti, Emilia ²

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     title = {Planes of matrices of constant rank and globally generated vector bundles},
     journal = {Annales de l'Institut Fourier},
     pages = {2069--2089},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Boralevi, Ada; Mezzetti, Emilia. Planes of matrices of constant rank and globally generated vector bundles. Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2069-2089. doi : 10.5802/aif.2983. https://aif.centre-mersenne.org/articles/10.5802/aif.2983/

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