On invariant rank two vector bundles on P2

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Simone Marchesi
Universitat de Barcelona
https://orcid.org/0000-0003-4371-601X
Jean Vallès
Université de Pau et des Pays de l’Adour

In this paper we characterize the rank two vector bundles on P 2 which are invariant under the actions of the parabolic subgroups Gp := Stabp(PGL(3)) fixing a point in the projective plane, GL := StabL(PGL(3)) fixing a line, and when p ∈ L, the Borel subgroup B = Gp∩GL of PGL(3). Moreover, we prove that the geometrical configuration of the jumping locus induced by the invariance does not, on the other hand, characterize the invariance itself. Indeed, we find infinite families that are almost uniform but not almost homogeneous.

Paraules clau
invariant bundles, logarithmic sheaves

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Marchesi, Simone; Vallès, Jean. «On invariant rank two vector bundles on P2». Publicacions Matemàtiques, 2023, vol.VOL 67, núm. 1, p. 259-75, https://raco.cat/index.php/PublicacionsMatematiques/article/view/412635.
Referències
V. Ancona and G. Ottaviani, Unstable hyperplanes for Steiner bundles and multidimensional matrices, Adv. Geom. 1(2) (2001), 165–192. DOI: 10.1515/advg.2001.012

E. Ballico, Uniform vector bundles of rank (n + 1) on Pn, Tsukuba J. Math. 7(2) (1983), 215–226. DOI: 10.21099/tkbjm/1496159821

A. Dimca and G. Sticlaru, Nearly free divisors and rational cuspidal curves, Preprint (2015). arXiv:1505.00666

J.-M. Drezet, Exemples de fibres uniformes non homogènes sur Pn−, C. R. Acad. Sci. Paris S´er. A-B 291(2) (1980), A125–A128

G. Elencwajg, Les fibres uniformes de rang 3 sur P2(C) sont homogènes, Math. Ann. 231(3) (1977/78), 217–227. DOI: 10.1007/BF01420242

G. Elencwajg, Des fibres uniformes non homog`enes, Math. Ann. 239(2) (1979), 185–192. DOI: 10.1007/BF01420375

G. Elencwajg, A. Hirschowitz, and M. Schneider, Les fibres uniformes de rang au plus n sur Pn(C) sont ceux qu’on croit, in: “Vector Bundles and Differential Equations” (Proc. Conf., Nice, 1979), Progr. Math. Birkhäuser, Boston, Mass., 1980, pp. 37–63. DOI: 10.1007/978-1-4684-9415-0_3

P. Ellia, Sur les fibres uniformes de rang (n + 1) sur P n, M´em. Soc. Math. France (N.S.) 7 (1982), 60 pp. DOI: 10.24033/msmf.292

P. Ellia, On jumping lines of vector bundles on P n, Ann. Univ. Ferrara Sez. VII Sci. Mat. 63(1) (2017), 103–112. DOI: 10.1007/s11565-017-0282-2

T. Kaneyama, On equivariant vector bundles on an almost homogeneous variety, Nagoya Math. J. 57 (1975), 65–86. DOI: 10.1017/S0027763000016561

T. Kaneyama, Torus-equivariant vector bundles on projective spaces, Nagoya Math. J. 111 (1988), 25–40. DOI: 10.1017/S0027763000000982

S. Marchesi and R. M. Miro-Roig ´ , Uniform Steiner bundles, Ann. Inst. Fourier (Grenoble) 71(2) (2021), 447–472. DOI: 10.5802/aif.3403

S. Marchesi and J. Valles` , Nearly free curves and arrangements: a vector bundle point of view, Math. Proc. Cambridge Philos. Soc. 170(1) (2021), 51–74. DOI: 10.1017/S0305004119000318

M. Maruyama, Singularities of the curve of jumping lines of a vector bundle of rank 2 on P2 , in: “Algebraic Geometry” (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer, Berlin, 1983, pp. 370–411. DOI: 10.1007/BFb0099972

C. Okonek, M. Schneider, and H. Spindler, “Vector Bundles on Complex Projective Spaces”, Progress in Mathematics 3, Birkhäuser, Boston, Mass., 1980. DOI: 10.1007/978-1-4757-1460-9

P. Orlik and H. Terao, “Arrangements of Hyperplanes”, Grundlehren der mathematischen Wissenschaften 300, Springer-Verlag, Berlin, 1992. DOI: 10.1007/978-3-662-02772-1

E. Sato, Uniform vector bundles on a projective space, J. Math. Soc. Japan 28(1) (1976), 123–132. DOI: 10.2969/jmsj/02810123

R. L. E. Schwarzenberger, Vector bundles on the projective plane, Proc. London Math. Soc. (3) 11(1) (1961), 623–640. DOI: 10.1112/plms/s3-11.1.623

J. Valles` , Fibr´es de Schwarzenberger et coniques de droites sauteuses, Bull. Soc. Math. France 128(3) (2000), 433–449. DOI: 10.24033/bsmf.2376

A. Van de Ven, On uniform vector bundles, Math. Ann. 195 (1972), 245–248. DOI: 10.1007/BF01419597