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On the ampleness of the normal bundle of line congruences

Arrondo Esteban, Enrique and Bertolini, Marina and Turrini, Cristina (2011) On the ampleness of the normal bundle of line congruences. Forum Mathematicum, 23 (2). pp. 223-244. ISSN 0933-7741

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Abstract

In this paper we study the normal bundle of the embedding of subvarieties of dimension n - 1 in the Grassmann variety of lines in P(n). Making use of some results on the geometry of the focal loci of congruences ([4] and [5]), we give some criteria to decide whether the normal bundle of a congruence is ample or not. Finally we apply these criteria to the line congruences of small degree in P(3).


Item Type:Article
Uncontrolled Keywords:Normal bundle, ampleness, line congruence, Grassmannian.
Subjects:Sciences > Mathematics > Algebra
ID Code:14736
References:

[1] E. Arrondo, Line congruences of low order, Milan Journal of Mathematics (formerly

Rendiconti del Seminario Matematico e Fisico di Milano) 70 (2002), 223–243.

[2] E. Arrondo, M. Bertolini and C. Turrini, Classification of smooth congruences with

a fundamental curve, in: Projective Geometry with Applications, pp. 43–56, Lecture

Notes in Pure and Applied Mathematic 166, Marcel Dekker, 1994.

[3] E. Arrondo, M. Bertolini and C. Turrini, Congruences of small degree in G.1; 4/,

Comm. Algebra 26 (1998), 3249–3266.

[4] E. Arrondo, M. Bertolini and C. Turrini, A focus on focal surface, Asian J. of Math.

5 (2001), 535–560.

[5] E. Arrondo, M. Bertolini and C. Turrini, Focal loci in G.1;N/, Asian J. of Math. 9

(2005), 449–472.

[6] E. Arrondo and J. Caravantes, On the Picard group of low-codimension subvarieties,

Indiana U. Math. J. 58 (2009), 1023–1050.

[7] E. Arrondo and M. L. Fania, Evidence to subcanonicity of codimension two submanifolds

of G.1; 4/, Internat. J. Math. 17 (2006), 157–168.

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Math. France, 50, Société Mathématique de France, 1992, Supplément au Bulletin

de la Société Mathématique de France, tome 120, fascicule 3.

[9] E. Ballico, Normal bundle to curves in quadrics, Bull. Soc. Math. France 109 (1981),

227–235.

[10] N. Goldstein, A special surface in the 4-quadric, Duke Math. J. 50 (1983), 745–761.

[11] N. Goldstein, Examples of non-ample normal bundles, Comp. Math 51 (1984),

189–192.

[12] N. Goldstein, Scroll surfaces in Gr.1; 3/, Rend. Sem. Mat. Univers. Politecn. Torino,

Special issue (1987), 69–75.

[13] R. Hartshorne, Algebraic Geometry,[1] E. Arrondo, Line congruences of low order, Milan Journal of Mathematics (formerly

Rendiconti del Seminario Matematico e Fisico di Milano) 70 (2002), 223–243.

[2] E. Arrondo, M. Bertolini and C. Turrini, Classification of smooth congruences with

a fundamental curve, in: Projective Geometry with Applications, pp. 43–56, Lecture

Notes in Pure and Applied Mathematic 166, Marcel Dekker, 1994.

[3] E. Arrondo, M. Bertolini and C. Turrini, Congruences of small degree in G.1; 4/,

Comm. Algebra 26 (1998), 3249–3266.

[4] E. Arrondo, M. Bertolini and C. Turrini, A focus on focal surface, Asian J. of Math.

5 (2001), 535–560.

[5] E. Arrondo, M. Bertolini and C. Turrini, Focal loci in G.1;N/, Asian J. of Math. 9

(2005), 449–472.

[6] E. Arrondo and J. Caravantes, On the Picard group of low-codimension subvarieties,

Indiana U. Math. J. 58 (2009), 1023–1050.

[7] E. Arrondo and M. L. Fania, Evidence to subcanonicity of codimension two submanifolds

of G.1; 4/, Internat. J. Math. 17 (2006), 157–168.

[8] E. Arrondo and I. Sols, On congruences of lines in the projective space, Mém. Soc.

Math. France, 50, Société Mathématique de France, 1992, Supplément au Bulletin

de la Société Mathématique de France, tome 120, fascicule 3.

[9] E. Ballico, Normal bundle to curves in quadrics, Bull. Soc. Math. France 109 (1981),

227–235.

[10] N. Goldstein, A special surface in the 4-quadric, Duke Math. J. 50 (1983), 745–761.

[11] N. Goldstein, Examples of non-ample normal bundles, Comp. Math 51 (1984),

189–192.

[12] N. Goldstein, Scroll surfaces in Gr.1; 3/, Rend. Sem. Mat. Univers. Politecn. Torino,

Special issue (1987), 69–75.

[13] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-

Verlag, New York, Heidelberg, Berlin, 1977.

[14] A. Papantonopoulou, Surfaces in the Grassmann variety G.1; 3/, Proc. Amer. Math.

Soc. 77 (1979), 15–18.

[15] M. Schneider and J. Zintl, The theorem of Barth-Lefschetz as a consequence of

Le Potier’s vanishing theorem, Manuscripta Math. 80 (1993), 259–263.

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