×

zbMATH — the first resource for mathematics

Arrangements of hyperplanes and vector bundles on \(\mathbb{P}^ n\). (English) Zbl 0804.14007
Let \({\mathcal H} = \{H_ 1, \dots, H_ m\}\) be an arrangement of \(m\) hyperplanes in a \(n\)-dimensional projective space \(\mathbb{P}^ n\). The authors consider the divisor \(D = H_ 1 \cup \cdots \cup H_ m\) and the locally free sheaf (or bundle) \(\Omega^ 1 (\log D)\) of the differential 1-forms with logarithmic poles on \(D\). Main result: If \(m \geq 2n + 3\), then the arrangement of hyperplanes \({\mathcal H}\) can be recovered from the sheaf \(\Omega^ 1 (\log D)\) unless all hyperplanes are osculating the same rational normal curve of degree \(n\).
In case \(n=2\) the variety of jumping lines of \(\Omega^ 1 (\log D)\) is a curve in the dual \(\mathbb{P}^ 2\) and the whole construction provides an interesting procedure that associates to a finite set of points in a plane a well determined curve through them.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves I , Grundlehren Math. Wiss., vol. 267, Springer-Verlag, New York, 1985. · Zbl 0559.14017
[2] H. F. Baker, Principles of Geometry, Vol. 3 , Cambridge Univ. Press, Cambridge, 1927. · JFM 53.0060.07
[3] W. Barth, Moduli of vector bundles on the projective plane , Invent. Math. 42 (1977), 63-91. · Zbl 0386.14005
[4] G. Bohnhorst and H. Spindler, The stability of certain vector bundles on \({\mathbf P}^ n\) , Complex algebraic varieties (Bayreuth, 1990), Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 39-50. · Zbl 0773.14008
[5] A. B. Coble, Associated sets of points , Trans. Amer. Math. Soc. 24 (1922), no. 1, 1-20. · JFM 49.0490.01
[6] A. B. Coble, Algebraic Geometry and Theta Functions , Amer. Math. Soc. Colloq. Publ., vol. 10, Amer. Math. Soc., New York, 1929. · JFM 55.0808.02
[7] H. Crapo and G. C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries , MIT Press, Cambridge, Mass.-London, 1970. · Zbl 0216.02101
[8] P. Deligne, Théorie de Hodge II , Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-57. · Zbl 0219.14007
[9] I. Dolgachev and M. Kapranov, Schur quadrics, cubic surfaces and rank \(2\) vector bundles over the projective plane , preprint 036-93, Math. Sci. Res. Inst., Berkeley, 1993. · Zbl 0809.14011
[10] I. Dolgachev and D. Ortland, Point Sets in Projective Spaces and Theta Functions , Astérisque (1988), no. 165, 210 pp. (1989), Soc. Math. France, Paris. · Zbl 0685.14029
[11] L. Ein, Normal sheaves of linear systems on curves , Algebraic Geometry: Sundance 1988, Contemp. Math., vol. 116, Amer. Math. Soc., Providence, 1991, pp. 9-18. · Zbl 0754.14007
[12] I. M. Gelfand and M. I. Graev, A duality theorem for general hypergeometric functions , Soviet Math. Dokl. 34 (1987), 9-13. · Zbl 0619.33006
[13] P. Griffiths and J. Harris, Principles of Algebraic Geometry , Pure Appl. Math., John Wiley & Sons, New York, 1978. · Zbl 0408.14001
[14] R. Hartshorne, Algebraic Geometry , Graduate Texts in Math., vol. 52, Springer-Verlag, New York, 1977. · Zbl 0367.14001
[15] K. Hulek, Stable rank-\(2\) vector bundles on \(\mathbb{P}_{2}\) with \(c_{1}\) odd , Math. Ann. 242 (1979), no. 3, 241-266. · Zbl 0407.32013
[16] M. M. Kapranov, Chow quotients of Grassmannians I , to appear in Adv. in Soviet Math. (I. M. Gelfand Festschrift), Amer. Math. Soc, 1993. · Zbl 0811.14043
[17] J. Le Potier, Fibrés stables de rang \(2\) sur \({\mathbf P}_{2}({\mathbf C})\) , Math. Ann. 241 (1979), no. 3, 217-256. · Zbl 0405.14008
[18] M. Maruyama, Elementary transformations in the theory of algebraic vector bundles , Algebraic geometry (La Rábida, 1981), Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 241-266. · Zbl 0505.14009
[19] M. Maruyama, Singularities of the curve of jumping lines of a vector bundle of rank \(2\) on \({\mathbf P}^{2}\) , Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 370-411. · Zbl 0529.14010
[20] D. Montesano, Su di un complesso di rette di terzo grado , Mem. Accad. Sci. Inst. Bologna (5) 3 (1893), 549-577. · JFM 25.1294.01
[21] C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces , Progress in Math., vol. 3, Birkhäuser, Boston, 1980. · Zbl 0438.32016
[22] Th. Reye, Ueber die allgemeine Fläche dritter Ordnung , Math. Ann. 55 (1901), 257-264. · JFM 32.0638.01
[23] T. G. Room, The Geometry of Determinantal Loci , Cambridge Univ. Press, Cambridge, 1937. · Zbl 0020.05402
[24] F. Schur, Ueber die durch collineare Grundgebilde erzeugten Curven and Flächen , Math. Ann. 18 (1881), 1-32. · JFM 13.0490.01
[25] R. L. E. Schwarzenberger, Vector bundles on the projective plane , Proc. London Math. Soc. (3) 11 (1961), 623-640. · Zbl 0212.26004
[26] R. L. E. Schwarzenberger, The secant bundle of a projective variety , Proc. London Math. Soc. (3) 14 (1964), 369-384. · Zbl 0123.38201
[27] A. N. Tjurin, On the classification of two-dimensional fibre bundles over an algebraic curve of arbitrary genus , Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 21-52, in Russian. · Zbl 0123.38103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.