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Two algebraic deformations of a K3 surface

  • Daniel Comenetz (a1)
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Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3 .

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COPYRIGHT: © Editorial Board of Nagoya Mathematical Journal 1981
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[1] Artin, M., Algebraic construction of Brieskorn’s resolutions, Journal of Algebra 29 (1974), 330–348.
[2] Atiyah, M. F., On analytic surfaces with double points, Proc. Roy. Soc. A247 (1958), 237–244.
[3] Burns, D. Jr. and Rapoport, M., On the Torelli problem for Kählerian K3 surfaces, Ann. scient. Éc. Norm. Sup, 4e série, t. 8 (1975), 235–274.
[4] Dolgachev, I. V., On special algebraic KS surfaces, Math. USSR Izvestia vol. 7, No. 4 (1973), 833–846.
[5] Goodman, J. E., Affine open subsets of algebraic varieties and ample divisors, Ann. of Math. 89 (1969), 160–183.
[6] Grothendieck, A. and Dieudonne, J., Éléments de Géométrie Algébrique, Publ. Math. IHES, nos. 4,.. (1960ff).
[7] Grothendieck, A. and Murre, J. P., The tame fundamental group of a formal neighborhood of a divisor with normal crossings on a scheme, Springer-Verlag, Lecture Notes in Math #208, Berlin (1971).
[8] Horikawa, E., On deformations of Quintic Surfaces, Inventiones math. 31 (1975), 43–85.
[9] Horikawa, E., Algebraic surfaces of general type with small C12 , II, Inventiones math. 37 (1976), 121–155.
[10] Kodaira, K., A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. 75 (1962), 146–162.
[11] Kodaira, K., On stability of compact submanifolds of complex manifolds, Amer. J. Math. 85 (1963), 79–94.
[12] Lascu, A. T., Sous-variété régulièrement contractibles d’une variété algébrique, Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 675–695.
[13] Matsusaka, T., Algebraic deformations of polarized varieties, Nagoya Math. J. 31 (1968), 185–245.
[14] Matsusaka, T., On stability of polarization, Number Theory, Alg. Geom. and Comm. Alg., in honor of Akizuki, Y., Tokyo (1973), 495–509.
[15] Matsusaka, T. and Mumford, D., Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. S6 (1964), 668–684.
[16] Mayer, A., Families of K3 surfaces, Nagoya Math. J. 48 (1972), 1–17.
[17] Mumford, D., Introduction to algebraic geometry, Harvard U. Notes (1965).
[18] Mumford, D., Algebraic geometry I, Springer-Verlag, Grundlehren der Mathematischen Wissenschaften 221, Berlin (1976).
[19] Reid, M., Hyperelliptic linear systems on a K3 surface, J. London Math. Soc. (2) 13 (1976), 427–437.
[20] Saint-Donat, B., Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602–639.
[21] Shafarevich, I. R., Basic algebraic geometry, Springer-Verlag, Grundlehren der Mathematischen Wissenschaften 213, Berlin (1974).
[22] Shafarevich, I. R. et al., Algebraic surfaces, Proc. Steklov Inst. Math. 75 (1965).
[23] Samuel, P., Méthodes d’algèbre abstraite en géométrie algébrique, seconde édition, Springer-Verlag, Ergebnisse der Math. 4, Berlin (1967).
[24] Serre, J.-P., Algèbre locale—multiplicités, Springer-Verlag, Lecture Notes in Math. 11, Berlin (1965).
[25] Wavrik, J. J., Deformations of branched coverings of complex manifolds, Amer. J. Math. 90 (1968), 926–960.
[26] Weil, A., Foundations of Algebraic Geometry, revised edition, Amer. Math. Soc. Publ. 29 (1960).
[27] Zariski, O., Foundations of a general theory of birational correspondences, Trans. Amer. Math. Soc. 53 (1943), 490–542 (+Collected Works, vol. 1, MIT Press).
[28] Zariski, O., Generalized semi-local rings, Summa Brasiliensis Math. vol. 1, fasc. 8, (1946), 169–195 (+ Coll. Works, vol. 2, MIT Press).
[29] Zariski, O., Introduction to the problem of minimal models in the theory of algebraic surfaces, Publ. Math. Soc. Japan 4 (1958) (+Coll. Works, vol. 2, MIT press).
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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