Secants, Bitangents, and Their Congruences

  • Kathlén Kohn
  • Bernt Ivar Utstøl NødlandEmail author
  • Paolo Tripoli
Part of the Fields Institute Communications book series (FIC, volume 80)


A congruence is a surface in the Grassmannian \(\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})\) of lines in projective 3-space. To a space curve C, we associate the Chow hypersurface in \(\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})\) consisting of all lines which intersect C. We compute the singular locus of this hypersurface, which contains the congruence of all secants to C. A surface S in \(\mathbb{P}^{3}\) defines the Hurwitz hypersurface in \(\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})\) of all lines which are tangent to S. We show that its singular locus has two components for general enough S: the congruence of bitangents and the congruence of inflectional tangents. We give new proofs for the bidegrees of the secant, bitangent and inflectional congruences, using geometric techniques such as duality, polar loci and projections. We also study the singularities of these congruences.

MSC 2010 codes:

14M15 14H50 14J70 14N10 14N15 51N35 14C17 



This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. We thank Daniele Agostini, Enrique Arrondo, Peter Bürgisser, Diane Maclagan, Emilia Mezzetti, Ragni Piene, Jenia Tevelev, and the anonymous referees for helpful discussions, suggestions and hints. Kathlén Kohn was supported by a Fellowship from the Einstein Foundation Berlin, Bernt Ivar Utstøl Nødland was supported by NRC project 144013, and Paolo Tripoli was supported by the University of Warwick and by EPSRC grant EP/L505110/1.


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Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Kathlén Kohn
    • 1
  • Bernt Ivar Utstøl Nødland
    • 2
    Email author
  • Paolo Tripoli
    • 3
  1. 1.Institute of MathematicsTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsUniversity of OsloOsloNorway
  3. 3.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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