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Projective surfaces with K-very ample line bundles of degree ≤ 4K + 4

Published online by Cambridge University Press:  22 January 2016

Edoardo Ballico  and
Andrew J. Sommese
Edoardo Ballico
Affiliation:
Dipartimento di Matematica, Università di Trento, 38050 Povo (TN), Italia, ballico@itncisca.bitnet, fax: Italy +461881624
Andrew J. Sommese
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, U.S.A., sommese.1@nd.edu, fax: U.S.A. +219-631-6579
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A line bundle, L, on a smooth, connected projective surface, S, is defined [7] to be k-very ample for a non-negative integer, k, if given any 0-dimensional sub-scheme with length , it follows that the restriction map is onto. L is 1-very ample (respectively 0-very ample) if and only if L is very ample (respectively spanned at all points by global sections). For a smooth surface, S, embedded in projective space by | L | where L is very ample, L being k-very ample is equivalent to there being no k-secant Pk−1 to S containing ≥ k + 1 points of S.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 136 , December 1994 , pp. 57 - 79
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1994

References

[1] Andreatta, M., “Surfaces of sectional genus ≤ 8 with no trisecant lines,” Archiv der Math., 60 (1993), 85–95.Google Scholar
[2] Arbarello, E., Cornalba, M., Griffiths, P. A., and Harris, J., Geometry of Algebraic Curves, Volume I, Grund. der math. Wissen., 267, Springer-Verlag, (1985).Google Scholar
[3] Ballico, E., “A characterization of the Veronese surface,” Proc. A. M. S., 105, (1989), 531–534.CrossRefGoogle Scholar
[4] Ballico, E., “On k-spanned projective surfaces,” 1988 L’Aquila Proceedings: Hyperplane sections, edited by Sommese, A. J., Biancofiore, A., and Livorni, E. L., Lecture Notes in Math., 1417 (1990), page 23, Springer-Verlag.Google Scholar
[5] Ballico, E., “On k-spanned embeddings of projective manifolds,” Manuscripta. Math., 75 (1992), 103–107.CrossRefGoogle Scholar
[6] Beltrametti, M. C. and Sommese, A. J., “On k-spannedness for projective surfaces,” 1988 L’Aquila Proceedings: Hyperplane sections, Lecture Notes in Math., 1417 (1990), 24–51, Springer-Verlag.Google Scholar
[7] Beltrametti, M. C. and Sommese, A. J., “Zero cycles and kth order embeddings of smooth projective surfaces,” 1988 Cortona Proceedings: Projective Surfaces and their Classification, ed. by Catanese, F., Symposia Mathematica, 32, (1992) INDAM, 33-48, Academic Press.Google Scholar
[8] Beltrametti, M. C. and Sommese, A. J., “On the preservation of k-very ampleness under adjunction,” Math. Z., 212 (1993), 257–284.Google Scholar
[9] Coppens, M. and Martens, G., “Secant spaces and Clifford’s theorem,” Compos, Math., 78 (1991), 193–212.Google Scholar
[10] Fulton, W. and Lazarsfeld, R., “On the connectedness of the degeneracy loci and special divisors,” Acta Math., 146 (1981), 271–283.Google Scholar
[11] Harris, J., “A bound on the geometric genus of projective varieties,” Ann. Norm. Super. Pisa, C. Sei. Ser. 4, 8 (1981), 35–68.Google Scholar
[12] Hartshorne, R., Algebraic Geometry, GTM 52, Springer-Verlag, New York (1987).Google Scholar
[13] Katsura, T. and Ueno, K., “Elliptic surfaces in characteristic p ,” Math. Ann., 272 (1985), 291–330.CrossRefGoogle Scholar
[14] Barz, P. Le, “Formules pour les trisécantes des surfaces algébriques,” L’Enseign. Math. II. Sér. 33 (1987), 1–66.Google Scholar
[15] Mella, M. and Palleschi, M., “The k-very ampleness of an elliptic quasi-bundle,” Abh. Math. Sem. Univ. Hamburg, 63 (1993), 215–226.Google Scholar
[16] Serrano, F., “Elliptic surfaces with an ample divisor of genus two,” Pac. J. Math. 152 (1992), 187–199.CrossRefGoogle Scholar
[17] Sommese, A. J., “The projective classification of surfaces, I: The adjunction mapping,” Duke Math. J., 46 (1979), 377–401.CrossRefGoogle Scholar
[18] Sommese, A. J. and Ven, A. Van de, “On the adjunction mapping,” Math. Ann., 278 (1987), 593–603.Google Scholar