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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Geometric properties of the double-point divisor

Author(s): 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.

References:

[ACGH]
E. Arbarello, M. Cornalba, P.A. Griffiths, and J.Harris, Geometry of algebraic curves, vol. 1, Springer-Verlag, New York, 1984. MR 86h:14019

[BM]
D. Bayer and D. Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, Cambridge Univ. Press, 1993, pp. 1-48. MR 95d:13032

[Ein]
L.Ein, The ramification divisor for branched coverings of $\mathbf{P}^{n}$, Math. Ann. 261 (1982), 483-485. MR 84c:14009

[EH]
D. Eisenbud and J. Harris, On varieties of minimal degree (a centennial account), Proceedings of the Symposia in Pure Math, vol 46, part 1, A.M.S, 1987, pp. 3-13. MR 89f:14042

[FL]
W. Fulton and R. Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, IL, 1980), Lecture Notes in Math., vol. 862, Springer-Verlag, New York, 1981, pp. 26-92. MR 83i:14002

[Ha1]
J. Harris, A bound on the geometric genus of projective varieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Ser. 4 8 (1981), 35-68. MR 82h:14010

[Ha2]
Joe Harris, Algebraic Geometry-a First Course, Graduate Texts in Math. Vol. 133, Springer-Verlag, New York, 1992. MR 93j:14001

[Hart]
Robin Hartshorne, Algebraic Geometry, Graduate Texts in Math. Vol. 52, Springer-Verlag, New York, 1977. MR 57:3116

[HOV]
K. Hulek, C. Okonek, and A. Van de Ven, Multiplicity-2 structures on Castelnuovo surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Ser. 4 13 (1986), 427-448. MR 88c:14052

[I]
Bo Ilic, Geometric properties of the double-point divisor, Thesis, Columbia University, 1995, E-prints: alg-geom/950309.

[Io]
Paltin Ionescu, Embedded projective varieties of small invariants, Algebraic Geometry, Bucharest 1982, Lecture Notes in Math., vol. 1056, Springer-Verlag, 1984, pp. 142-186. MR 85m:14024

[Kl]
Steven L. Kleiman, The enumerative theory of singularities, Real and Complex Singularities, Oslo 1976, P. Holm (ed.), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 297-396. MR 58:27960

[Lar]
M. E. Larsen, On the topology of complex projective manifolds, Invent. Math. 19 (1973), 251-260. MR 47:7058

[Mum]
David Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math., vol. 1358, Springer-Verlag, Berlin, 1988. MR 89k:14001

[Ro]
Leonard Roth, On the projective classification of surfaces, Proc. London Math. Soc. (2) 42 (1937), 142-170.

[So]
Andrew John Sommese, Hyperplane sections of projective surfaces. I: The adjunction mapping, Duke Math. Journal 46 (1979), 377-401. MR 82f:14033

[V]
E. Viehweg, Vanishing theorems, Jour. Reine Angew. Math. 335 (1982), 1-8. MR 83m:14011

[Zak]
F. L. Zak, Tangents and secants of algebraic varieties, Translations of Math. Monographs vol. 127, American Math. Society, 1993. MR 94i:14053

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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
Email: ilic@math.ucla.edu

DOI: 10.1090/S0002-9947-98-01928-X
PII: S 0002-9947(98)01928-X
Keywords: Double-point divisor, Roth variety, Castelnuovo variety, secant variety, conductor, projection
Received by editor(s): July 20, 1996
Copyright of article: Copyright 1998, American Mathematical Society




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