Arrondo Esteban, Enrique and Bertolini, Marina and Turrini, Cristina
(2005)
*Focal loci in G(1,N).*
Asian journal of mathematics, 9
(4).
pp. 449-472.
ISSN 1093-6106

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## Abstract

We introduce the different focal loci (focal points, planes and hyperplanes) of (n - 1)-dimensional families (congruences) of lines in P-n and study their invariants, geometry and the relation among them. We also study some particular congruences whose focal loci have special behaviour, namely (n - 1)-secant lines to an (n - 2)-fold and (n - 1)-tangent lines to a hypersurface. In case n = 4 we also give, under some smoothness assumptions, a classification result for these congruences.

Item Type: | Article |
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Uncontrolled Keywords: | Focal locus; congruence; Grassmannian of lines |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 14822 |

References: | [1] E. Arrondo, Projections on Grassmannians of lines and characterization of Veronese vari- eties, J. Algebraic Geom., 8:50 (1999), pp. 85–101. [2] E. Arrondo, Line congruences of low order, Milan Journal of Math., 70 (2002), pp. 223–243. [3] E. Arrondo, M. Bertolini and C. Turrini, Classification of smooth congruences with a fundamental curve, Projective Geometry with applications, Number 166 in LN. Marcel Dekker, 1994. [4] E. Arrondo, M. Bertolini and C. Turrini, A focus on focal surfaces, Asian J. of Math., 5:3 (2001), pp. 535–560. [5] M. Bertolini and C. Turrini, Surfaces in P4 with no quadrisecant lines, Beitrage zur Algebra und Geometrie, 39 (1998), pp. 31–36. [6] C. Ciliberto and E. Sernesi, Singularities of the theta divisor and congruences of planes, Journal of Alg. Geom., 1:2 (1992), pp. 231–250. [7] N. Goldstein, The geometry of surfaces in the 4-quadric, Rend. Sem. Mat. Univers. Politecn. Torino, 43:3 (1985), pp. 467–499. [8] R. Hartshorne, Algebraic Geometry, Number 52 in GTM. Springer Verlag, New York - Hei- delberg - Berlin, 1977. [9] S. Katz and S.A. Strømme, schubert, a Maple package for intersection theory, available at http://www.mi.uib.no/schubert/. [10] E.L. Livorni, On the existence of some surfaces, Algebraic Geometry Proc., Number 1417 Springer Verlag, New York - Heidelberg - Berlin, 1977, pp. 155–179. [11] P. Le Barz, Validit´e de certaines formules de g´eom´etrie enumerative, C. R. Acad. Sc. Paris, 289 (1979), pp. 755–758. [12] P. Le Barz, Formules pour les trisecantes des surfaces alg´ebriques, L’Enseignement Math´ematique, 33 (1987), pp. 1–66. |

Deposited On: | 18 Apr 2012 09:33 |

Last Modified: | 06 Feb 2014 10:10 |

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