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Kleppe, Jan O.; Ottem, John C. (2012)
Publisher: World Scientific Publishing
Languages: English
Types: Article
Subjects: Cubic surfaces, Space curves, Hilbert flag-scheme, VDP::Matematikk og Naturvitenskap: 400::Matematikk: 410, Mathematics - Algebraic Geometry, Hilbert scheme, Quartic surfaces, 14C05 (Primary), 14C20, 14K30, 14J28, 14H50 (Secondary), :Matematikk og Naturvitenskap: 400::Matematikk: 410 [VDP]
We study maximal families W of the Hilbert scheme, H(d, g)sc, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d, g)sc and we determine dim W. If s = 4 and W is an irreducible component of H(d, g)sc, then the Picard number of S is at most 2 and we explicitly describe, also for s ≥ 5, non-reduced and generically smooth components in the case Pic(S) is generated by the classes of a line and a smooth plane curve of degree s - 1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d, g)sc, thus making progress in proving a conjecture for such families. Electronic version of an article published as Kleppe, J. O., & Ottem, J. C. (2015). Components of the Hilbert scheme of space curves on low-degree smooth surfaces. International Journal of Mathematics, 26(02), 1550017. © World Scientific Publishing Company.
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    • [1] A. Dolcetti, G. Pareschi. On Linearly Normal Space Curves. Math. Z. 198, (1988), 73-82.
    • [2] D. Eklund Curves on Heisenberg invariant quartic surfaces in projective 3-space. Preprint; arXiv:1010.4058
    • [3] Ph. Ellia: D'autres composantes non réduites de Hilb P3, Math. Ann. 277, (1987) 433-446.
    • [4] G. Ellingsrud. Sur le schéma de Hilbert des variétés de codimension 2 dans Pe a cône de Cohen-Macaulay, Ann. Scient. Éc. Norm. Sup. 8 (1975), 423-432.
    • [5] G. Ellingsrud and C. Peskine. Anneau de Gorenstein associé à un fibré inversible sur une surface de l'espace et lieu de Noether-Lefschetz Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), 29-42, Hindustan Book Agency, Delhi, 1993.
    • [6] G. Fløystad. Determining obstructions for space curves, with application to non-reduced components of the Hilbert scheme. J. reine angew. Math. 439 (1993), 11-44.
    • [7] A. Grothendieck. Les schémas de Hilbert. Séminaire Bourbaki, exp. 221 (1960).
    • [8] A. Grothendieck. Les schémas des Picard: théorème d'existence. Séminaire Bourbaki, exp. 232 (1961/62).
    • [9] S. Giuffrida. Graded Betti numbers and Rao modules of curves lying on a smooth cubic surface in P3, Queen's Papers in Pure and Applied Math. 88 (1991) A1-A61.
    • [10] S. Giuffrida R. Maggioni. Generators for the ideal of an integral curve lying on a smooth quartic surface, J. Pure and Applied algebra 76 (1991) 317-332.
    • [11] P. Griffiths, J. Harris. Infinitesimal variations of Hodge structure (II). An infinitesimal invariant of Hodge classes. Comp. Math. 50 (1983), 207-265.
    • [12] L. Gruson, Chr. Peskine. Genre des courbes de l'espace projectif, In: Proc. Tromsø 1977, Lectures Notes in Math., 687 (1978), Springer-Verlag, New York, 1986.
    • [13] L. Gruson, Chr. Peskine. Genre des courbes de l'espace projectif (II), Ann. Sci. Éc. Norm. Sup. (4), 15 (1982), 401-418.
    • [14] R. Hartshorne. Algebraic Geometry. Graduate Texts in Math. Vol. 52 Springer-Verlag, New York, 1983.
    • [15] R. Hartshorne. Deformation theory. Graduate Texts in Math. Springer-Verlag, New York, 2010.
    • [16] R. Hartshorne. Connectedness of the Hilbert scheme. Publ. Math. I.H.E.S. 29 (1966), 5-48.
    • [17] J.O. Kleppe. Non-reduced components of the Hilbert scheme of smooth space curves, in Proceedings Rocca di Papa 1985 Lectures Notes in Math. Vol. 1266 Springer-Verlag (1987).
    • [18] J.O. Kleppe. The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in 3-space. Preprint (part of thesis), March 1981, Univ. of Oslo. http://www.iu.hio.no/∼jank/papers.htm.
    • [19] J.O. Kleppe. Liaison of families of subschemes in Pn, in “Algebraic Curves and Projective Geometry, Proceedings (Trento, 1988),” Lectures Notes in Math. Vol. 1389 Springer-Verlag (1989).
    • [20] J.O. Kleppe. The Hilbert scheme of space curves of small Rao module with an appendix on non-reduced components. Preprint June 1996. http://www.iu.hio.no/∼jank/papers.htm.
    • [21] J.O. Kleppe. On the existence of Nice Components in the Hilbert Scheme H(d, g) of Smooth Connected Space Curves. Boll. U.M.I (7) 8-B (1994), 305-326.
    • [22] J.O. Kleppe. The Hilbert Scheme of Space Curves of small diameter. Annales de l'institut Fourier 56 no. 5 (2006), p. 1297-1335.
    • [23] A. Laudal. Formal Moduli of Algebraic Structures. Lectures Notes in Math., Vol. 754, Springer-Verlag, New York, 1979.
    • [24] A. Lopez. Noether-Lefschetz theory and the Picard group of Surfaces. Memoire Amer. Math. Soc. 89 (1991).
    • [25] M. Martin-Deschamps, D. Perrin. Sur la classification des courbes gauches, Asterisque, 184-185 (1990).
    • [26] M. Martin-Deschamps, D. Perrin Courbes Gauches et Modules de Rao, J. reine angew. Math. 439 (1993), 103-145.
    • [27] M. Martin-Deschamps, D. Perrin. Le schéma de Hilbert des Courbes Gauches localement CohenMacaulay n'est (presque) jamais reduit. Ann. Scient. Éc. Norm. Sup. 4 t. 29 (1996), 757-785.
    • [28] S. Mori author. On degrees and genera of curves on smooth quartic surfaces in P3. Nagoya Math. J. 96, (1984) 127-132.
    • [29] D. Mumford. Further pathologies in algebraic geometry, Amer. J. Math., 84, 1962, 642-648.
    • [30] H. Nasu. Obstructions to deforming space curves and non-reduced components of the Hilbert scheme. Publ. Res. Inst. Math. Sci. 42 (2006), no 1, 117-141.
    • [31] V. Nikulin. Integral symmetric bilinear forms and some of their geometric applications. Math. USSR Izv. 14, 103-167, (1980)
    • [32] B. Saint-Donat. Projective models of K-3 surfaces. Amer. J. Math. 96, 1974.
    • [33] V.A. Iskovskikh, I.R. Shafarevich, Algebraic Geometry II: Algebraic Surfaces, Encyclopaedia of Mathematical Sciences, Vol. 35, Springer-Verlag (1996).
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