Publication:
A focus on focal surfaces

Loading...
Thumbnail Image
Official URL
Full text at PDC
Publication Date
2001
Authors
Bertolini, Marina
Turrini, Cristina
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
International Press
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
Congruences of lines in P3, i.e. two-dimensional families of lines, and their focal surfaces, have been a popular object of study in classical algebraic geometry. They have been considered recently by several authors as Arrondo, Goldstein, Sols, Verra. Aim of the paper under review is to study from the modern point of view the notions appearing in this context, so to prove in a rigorous way some classical results. More precisely, a congruence is a surface X G(1, 3), its focal locus F, in general a surface, is the branch locus of the natural projection qX from the incidence correspondence IX to P3. It results that a general line of X contains two foci counted with multiplicity, a line L whose points are all foci is called a focal line. In the paper under review the authors compute several invariants of the focal surface of a smooth congruence, e.g. its degree, class and sectional genus, and the degree of its nodal and cuspidal curves (assuming that these are the only 1-dimensional components of its singular locus). Then they study in detail some important examples of congruences, i.e. the bisecant lines of a smooth curve C in P3, the bitangents and the inflectional tangents of a smooth surface . In the first case they find that in general the focal surface F is all formed by focal lines, which are the stationary bisecants. In the other examples they find, among other results, that is a component of the focal surface with high multiplicity, and that at least one component of F is formed by focal lines.Motivated by the examples they state a series of conjectures about congruences whose focal surface is not irreducible or not reduced, and about singularities of congruences of bitangents or inflectionary tangents to not necessarily smooth surfaces of P3.
Description
UCM subjects
Unesco subjects
Keywords
Citation
E. Arrondo – M. Gross, On smooth surfaces in Gr(1,P3) with a fundamental curve, Manuscripta Math., 79, (1993), 283-298. E. Arrondo – I. Sols – R. Speiser, Global moduli of contacts, Arkiv f¨or Math., 35(1997), 1-57. C. Ciliberto – E. Sernesi, Singularities of the theta divisor and congruences of planes, Journal of Alg. Geom., 1 no. 2 (1992), 231-250. G. Fano, Studio di alcuni sistemi di rette considerati come superficie dello spazio a cinque dimensioni, Annali di Matematica, 21 (1893), 141-192. N. Goldstein, The geometry of surfaces in the 4-quadric, Rend. Sem. Mat. Univers. Politecn. Torino, 43, 3 (1985), 467-499. P. Griffiths – J. Harris, Algebraic geometry and local differential geometry, Ann. Sci.´Ecole Norm. Sup. (4) 12 (1979), 355-452. M. Gross, The distribution of bidegrees of smooth surfaces in G(1, P3), Math. Ann. 292 (1992), 127-147. R. W. H. T. Hudson, Kummer’s quartic surface, Cambridge Univ. Press, ed. 1990. T. Johnsen, Plane projections of a smooth space curve, in “Parameter spaces”, Banach Center Publications, VOl. 36 (1996), 89-110. S. Katz, – S.A. Strømme, schubert, a Maple package for intersection theory, Available at http://www.math.okstaste.edu/∼katz/schubert.html or by anonymous ftp from ftp.math.okstate.edu or linus.mi.uib.no, cd pub/schubert. C. McCrory – T. Shifrin, Cusps of the projective Gauss map, J. Differential Geometry, 19 (1984), 257-276. C. McCrory – T. Shifrin – R. Varley, The Gauss map of a generic hypersurface in P4, J. Differential Geometry, 30 (1989), 689-759. C. Peskine – L. Szpiro, Liaison des vari´et´es alg´ebriques, I, Invent. Math. 26 (1974), 271-302. L. Roth, Line congruences in three dimensions, Proc. London Math. Soc. (2), 32(1931), 72-86. L. Roth, Some properties of line congruences, Proc. Camb. Phil. Soc., 27 (1931),190-200. G. Salmon, A treatise on the analytic geometry of three dimension, Vol. II, 5th ed. Chelsea Pub. Co., 1965. R. Schumacher, Classification der algebraischen Strahlensysteme, 37 (1890), 100-140. A. Verra, Geometria della retta in dimensione 2, unpublished paper (1986). R. J. Walker, Algebraic Curves, Reprint by Springer-Verlag, 1978. G. E.Welters, Abel-Jacobi isogenies for certain types of Fano threefolds, Mathematical Centre Tracts 141, Amsterdam 1981.
Collections