On the cohomological strata of families of vector bundles on algebraic surfaces

Ballico, Edoardo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 2 (1991), p. 155-165 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

In this Note we study certain natural subsets of the cohomological stratification of the moduli spaces of rank $2$ vector bundles on an algebraic surface. In the last section we consider the following problem: take a bundle $E$ given by an extension, how can one recognize that $E$ is a certain given bundle? The most interesting case considered here is the case $E = T P^{3} (t)$ since it applies to the study of codimension $1$ meromorphic foliations with singularities on $P^{3}$ .

In questa Nota si studiano sottoinsiemi opportuni della stratificazione per gruppi di coomologia degli schemi di moduli di fibrati vettoriali di rango $2$ sulle superfici algebriche. Inoltre si considera il seguente problema: dato un fibrato $E$ come estensione, come riconoscere che $E$ è isomorfo ad un altro fibrato assegnato? Si considera il caso $E = T P^{3} (t)$ che viene applicato allo studio delle fogliazioni meromorfe con singolarità su $P^{3}$ .

Publié le : 1991-06-01

MR 1120135 | Zbl 0745.14004

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     author = {Edoardo Ballico},
     title = {On the cohomological strata of families of vector bundles on algebraic surfaces},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {2},
     year = {1991},
     pages = {155-165},
     zbl = {0745.14004},
     mrnumber = {1120135},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1991_9_2_2_155_0}
}

Ballico, Edoardo. On the cohomological strata of families of vector bundles on algebraic surfaces. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 2 (1991) pp. 155-165. http://gdmltest.u-ga.fr/item/RLIN_1991_9_2_2_155_0/

Bibliographie

[1] Ballico, E., On the minimal free resolution of s points in $P^{3}$ . Rend. Circ. Mat. Palermo, s. 2, vol. 37, 1988, 431-443. | MR 1008205 | Zbl 0702.14008

[2] Ballico, E., Rank- $2$ vector bundles on a surface with many sections and low $c_{2}$ . Geom. Dedicata, vol. 29, # 9 , 109-124. | MR 989190 | Zbl 0685.14011

[3] Ballico, E., Life and death of a cohomology class. J. Algebra, to appear. | MR 1125695 | Zbl 0752.14017

[4] Ballico, E., On rank $2$ vector bundles on an algebraic surface. Preprint. | MR 1164095

[5] Ballico, E. - Brussee, R., On the unbalance of vector bundles on a blown-up surface. Preprint.

[6] Ballico, E. - Chiantini, L., Some properties of stable rank $2$ bundles on algebraic surfaces. Preprint. | MR 1166263 | Zbl 0757.14021

[7] Ballico, E. - Ellia, Ph., On space curves with maximal genus in the range $A$ . Rev. Roumaine Math. pures appl., vol. 33, 1988, 165-174. | MR 948157 | Zbl 0662.14011

[8] Ballico, E. - Geramita, A. V., The minimal free resolution of the ideal of $s$ general points in $P^{3}$ . Proceedings of the 1984 Vancouver conference in Algebraic Geometry, CMS Conference Proc, vol. 6, 1986, 1-10. | MR 846012 | Zbl 0621.14003

[9] Ballico, E. - Newstead, P., Uniform bundles on quadratic surfaces and some related varieties. J. London Math. Soc, s. 2, vol. 31, 1985, 211-223. | MR 809942 | Zbl 0535.14009

[10] Catanese, F., Footnotes to a theorem of I. Reider. Proceedings on Algebraic Geometry (L'Aquila. 1988), Lect. Notes in Math., 1417, Springer Verlag, 1989. | MR 1040551 | Zbl 0705.14015

[11] Ein, L., Stable vector bundles on projective spaces in char $p > 0$ . Math. Ann., vol. 254, 1980. 53-72. | MR 597836 | Zbl 0431.14003

[12] Ellingsrud, G., Sur le schéma de Hilbert des variétés de codimension $2$ dans $P^{e}$ a cone de Cohen-Macaulay. Ann. Scient. Ec. Norm. Sup., s. 4, vol. 8, 1975, 423-432. | MR 393020 | Zbl 0325.14002

[13] Geramita, A. V. - Maroscia, P., The ideal of forms vanishing at a finite set of points in $P^{n}$ . J. Algebra, vol. 90, 1984, 528-555. | MR 760027 | Zbl 0547.14001

[14] Gomez-Mont, X., Universal families of foliations by curves. Proceedings of the Conference on Dynamical Systems, Astérisque, vol. 150-151, 1987, 109-129. | MR 923596 | Zbl 0641.32014

[15] Gomez-Mont, X., Foliations by curves of complex analytic spaces. In: Contemporary Mathematics. vol. 58, part III, 1987, 123-141. | MR 893860 | Zbl 0614.32013

[16] Gomez-Mont, X., The transverse dynamics of a holomorphic flow. Ann. Math., vol. 127, 1988, 49-92. | MR 924673 | Zbl 0639.32013

[17] Gomez-Mont, X., Holomorphic foliations in ruled surfaces. Trans. Amer. Math. Soc, vol. 312, 1989, 179-201. | MR 983870 | Zbl 0669.57012

[18] Gomez-Mont, X. - Kempf, G., Stability of meromorphic vector fields in projective spaces. Comment. Math. Helvetici, vol. 64, 1989, 462-473. | MR 998859 | Zbl 0709.14008

[19] Gruson, L. - Peskine, C., Genre des courbes de l'espace projectif. Proceedings on Algebraic Geometry (Tromsø, 1977), Lect. Notes in Math., 687, Springer Verlag, 1978, 31-60. | MR 527229 | Zbl 0412.14011

[20] Hartshorne, R., Stable vector bundles of rank $2$ on $P^{3}$ . Math. Ann., vol. 238, 1978, 229-280. | MR 514430 | Zbl 0411.14002

[21] Hartshorne, R. - Hirschowitz, A., Cohomology of a general instanton bundle. Ann. Scient. Ec. Norm. Sup., s. 4, vol. 15, 1982, 365-390. | MR 683638 | Zbl 0509.14015

[22] Hirschowitz, A., Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles génériques. J. reine angew. Math., vol. 397, 1989, 208-213. | MR 993223 | Zbl 0686.14013

[23] Idá, M., On the homogeneous ideal of the generic union of lines in $P^{3}$ . J. reine angew. Math., vol. 403, 1990, 67-153. | MR 1030412 | Zbl 0681.14032

[24] Migliore, J. - Miro-Roig, R., On k-Buchsbaum curves in $P^{3}$ . Communications in Alg., vol. 18, 1990, 2403-2422. | MR 1074235 | Zbl 0716.14017

[25] Mumford, D., Lectures on curves on an algebraic surface. Princeton Univ. Press, New York1966. | MR 209285 | Zbl 0187.42701

[26] Okonek, C. - Schneider, M. - Spindler, H., Vector bundles on complex projective spaces. Birkhauser, Boston-Basel-Stuttgart 1980. | MR 561910 | Zbl 0438.32016

[27] Peskine, C. - Szpiro, L., Liaison des variétés algébriques. Invent. Math., vol. 26, 1974, 271-302. | MR 364271 | Zbl 0298.14022

[28] Sauer, T., Smoothing protectively Cohen-Macaulay space curves. Math. Ann., vol. 272, 1985, 83-90. | MR 794092 | Zbl 0546.14023

[29] Van De Ven, A., On uniform vector bundles. Math. Ann., vol. 195, 1972, 245-248. | MR 291182 | Zbl 0215.43202