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Geometric properties of the double-point divisor

Author: Bo Ilic
Journal: Trans. Amer. Math. Soc. 350 (1998), 1643-1661
MSC (1991): Primary 14N05, 14C20, 14J40
MathSciNet review: 1422899
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Abstract: The locus of double points obtained by projecting a variety $X^{n} \subset \mathbf {P}^N$ to a hypersurface in $\mathbf {P}^{n+1}$ moves in a linear system which is shown to be ample if and only if $X$ is not an isomorphic projection of a Roth variety. Such Roth varieties are shown to exist, and some of their geometric properties are determined.

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  • E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
  • Dave Bayer and David Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991) Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 1–48. MR 1253986
  • Lawrence Ein, The ramification divisors for branched coverings of ${\bf P}^{n}_{k}$, Math. Ann. 261 (1982), no. 4, 483–485. MR 682661, DOI
  • David Eisenbud and Joe Harris, On varieties of minimal degree (a centennial account), Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13. MR 927946, DOI
  • William Fulton and Robert Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 26–92. MR 644817
  • Joe Harris, A bound on the geometric genus of projective varieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 35–68. MR 616900
  • Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558
  • Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • K. Hulek, C. Okonek, and A. Van de Ven, Multiplicity-$2$ structures on Castelnuovo surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 3, 427–448. MR 881100
  • Bo Ilic, Geometric properties of the double-point divisor, Thesis, Columbia University, 1995, E-prints: alg-geom/950309.
  • Paltin Ionescu, Embedded projective varieties of small invariants, Algebraic geometry, Bucharest 1982 (Bucharest, 1982) Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984, pp. 142–186. MR 749942, DOI
  • Steven L. Kleiman, The enumerative theory of singularities, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 297–396. MR 0568897
  • Mogens Esrom Larsen, On the topology of complex projective manifolds, Invent. Math. 19 (1973), 251–260. MR 318511, DOI
  • David Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1988. MR 971985
  • Leonard Roth, On the projective classification of surfaces, Proc. London Math. Soc. (2) 42 (1937), 142–170.
  • Andrew John Sommese, Hyperplane sections of projective surfaces. I. The adjunction mapping, Duke Math. J. 46 (1979), no. 2, 377–401. MR 534057
  • Eckart Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1–8. MR 667459, DOI
  • F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494

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Additional Information

Bo Ilic
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90024

Keywords: Double-point divisor, Roth variety, Castelnuovo variety, secant variety, conductor, projection
Received by editor(s): July 20, 1996
Article copyright: © Copyright 1998 American Mathematical Society