Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-01T08:11:45.281Z Has data issue: false hasContentIssue false

Embeddings of curves and surfaces

Published online by Cambridge University Press:  22 January 2016

Fabrizio Catanese
Affiliation:
Mathematisches Institut der Universität, Bunsenstrasse 3-5, D-37073 Göttingen, Germany, catanese@cfgauss.uni-math.gwdg.de
Marco Franciosi
Affiliation:
Scuola Normale Superiore, piazza dei Cavalieri 7, I-56126 Pisa, Italy, francios@cibs.sns.it, francios@gauss.dm.unipi.it
Klaus Hulek
Affiliation:
Institut für Mathematik, Univ. Hannover, Postfach 6009, D-30060 Hannover, Germany, hulek@math.uni-hannover.de
Miles Reid
Affiliation:
Math Inst, Univ. of Warwick, Coventry CV4 7AL, England, Miles@Maths.Warwick.Ac.UK
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a general embedding theorem for Cohen-Macaulay curves (possibly nonreduced), and deduce a cheap proof of the standard results on pluricanonical embeddings of surfaces, assuming vanishing H1(2KX) = 0.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

References

[Ar1] Artin, M., Some numerical criteria for contractibility of curves on algebraic surfaces, Amer. J. Math., 84 (1962), 485496.CrossRefGoogle Scholar
[Ar2] Artin, M., On isolated rational singularities of surfaces, Amer. J. Math., 88 (1966), 129136.CrossRefGoogle Scholar
[B-P-V] Barth, W., Peters, C. and Ven, A. Van de, Compact complex surfaces, Springer, 1984.CrossRefGoogle Scholar
[Ba1] Bauer, I., Geometry of algebraic surfaces admitting an inner projection, Preprint Pisa no. 1.72, 708 (1992).Google Scholar
[Ba2] Bauer, I., Embeddings of curves, Manuscr. Math., 87 (1995), 2734.CrossRefGoogle Scholar
[Ba3] Bauer, I., Inner projections of algebraic surfaces: a finiteness result, J. reine angew. Math., 460 (1995), 113.Google Scholar
[Ba4] Bauer, I., The classification of surfaces in ℙ5 having few trisecants, Rend. del Sem. Mat. di Torino., 56/1 (1998).Google Scholar
[Bo] Bombieri, E., Canonical models of surfaces of general type, Publ. Math. IHES, 42 (1973), 171219.CrossRefGoogle Scholar
[B-M] Bombieri, E. and Mumford, D., Enriques’ classification of surfaces in char p II, in ‘Complex Analysis and Algebraic Geometry’, collected papers dedicated to Kodaira, K., Iwanami Shoten, Tokyo (1977), pp. 2342.CrossRefGoogle Scholar
[Ca1] Catanese, F., Pluricanonical Gorenstein curves, in ‘Enumerative Geometry and Classical Algebraic Geometry’, Nice, Prog. in Math. 24, Birkhäuser (1981), pp. 5195.Google Scholar
[Ca2] Catanese, F., Footnotes to a theorem of Reider, in ‘Algebraic Geometry’, Proceedings of the L’Aquila conference 1988. (Sommese, A. J., Biancofore, A., Livorni, E. L., eds.), Springer LNM 1417 (1990), pp. 6774.Google Scholar
[C-F] Catanese, F. and Franciosi, M., Divisors of small genus on algebraic surfaces and projective embeddings, Proceedings of the conference “Hirzebruch 65”, Tel Aviv 1993, Contemp. Math., A.M.S. (1994), subseries ‘Israel Mathematical Conference Proceedings’ Vol. 9 (1996), 109140.Google Scholar
[C-H] Catanese, F. and Hulek, K., Rational surfaces in ℙ4 containing a plane curves, Ann. Mat. Pura Appl. (4), 172 (1997), 229256.CrossRefGoogle Scholar
[C-C-M] Catanese, F., Ciliberto, C. and Lopes, M. Mendes, On the classification of irregular surfaces of general type with nonbirational bicanonical map, Trans. Amer. Math. Soc., 350 (1998), 275308.CrossRefGoogle Scholar
[C-F-M] Ciliberto, C., Francia, P. and Lopes, M. Mendes, Remarks on the bicanonical map for surfaces of general type, Math. Z., 224 (1997), 137166.CrossRefGoogle Scholar
[Ek] Ekedahl, T., Canonical models of surfaces of general type in positive characteristic, Publ. Math. IHES, 67 (1988), 97144.CrossRefGoogle Scholar
[F] Franciosi, M., On k-spanned surfaces of sectional genus 8, Publ. Dip. Mat., Univ. di Pisa 1.114, 846, 1995.Google Scholar
[Fr1] Francia, P., The bicanonical map for surfaces of general type, unpublished manuscript, c. 1982-1987.Google Scholar
[Fr2] Francia, P., On the base points of the bicanonical system, in ‘Problems in the theory of surfaces and their classification’ (Cortona, Oct. 1988) (Catanese, F. and others, eds.), Symp. Math. XXXII, Acad. Press INDAM, Rome (1991), pp. 141150.Google Scholar
[Gr-Ha] Grothendieck, A. (notes by Hartshorne, R.), Local cohomology, Springer, LNM 41, 1967.Google Scholar
[Ha] Hartshorne, R., Algebraic Geometry, Springer, 1977.CrossRefGoogle Scholar
[Mu] Mumford, D., The canonical ring of an algebraic surface, Ann. of Math., 76 (1962), 612615.Google Scholar
[ML] Lopes, M. Mendes, Adjoint systems on surfaces, Boll. Un. Mat. Ital. A (7), 10 (1996), 169179.Google Scholar
[Ra] Ranestad, K., Surfaces of degree 10 in the projective fourspace, in ‘Problems in the theory of surfaces and their classification’ (Cortona, Oct. 1988) (Catanese, F. and others, eds.), Symp. Math. XXXII, Acad. Press INDAM, Rome (1991), pp. 271307.Google Scholar
[Re] Reid, M., Nonnormal del Pezzo surfaces, Publications of RIMS 30:5 (1995), 695727.Google Scholar
[Rei] Reider, I., Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math., 127 (1988), 309316.CrossRefGoogle Scholar
[Se1] Serre, J.-P., Faisceaux algébriques cohérents, Ann. of Math., 61 (1955), 197278.CrossRefGoogle Scholar
[Se2] Serre, J.-P., Courbes algébriques et corps de classes, Hermann, Paris, 1959 (English translation, Springer, 1990).Google Scholar
[S-B] Shepherd-Barron, N. I., Unstable vector bundlesd and linear systems on surfaces in characteristic p, Invent. Math., 106 (1991), 243262.CrossRefGoogle Scholar