Planes of matrices of constant rank and globally generated vector bundles

Boralevi, Ada; Mezzetti, Emilia

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

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 $c_2\le \left(\genfrac{}{}{0pt}{}{c_1+1}{2}\right)$ 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 $c_1\le 3$.

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 $c_2\le \left(\genfrac{}{}{0pt}{}{c_1+1}{2}\right)$ 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 $c_1\le 3$.

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