The Multidegree of the Multi-Image Variety

  • Laura EscobarEmail author
  • Allen Knutson
Part of the Fields Institute Communications book series (FIC, volume 80)


The multi-image variety is a subvariety of \(\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})^{n}\) that parametrizes all of the possible images that can be taken by n fixed cameras. We compute its cohomology class in the cohomology ring of \(\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})^{n}\) and its multidegree as a subvariety of \((\mathbb{P}^{5})^{n}\) under the Plücker embedding.

MSC 2010 codes:




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 for Research in Mathematical Sciences. The authors would like to thank their anonymous referees as well as Jenna Rajchgot and Bernd Sturmfels. The first author was supported by the Fields Institute for Research in Mathematical Sciences.


  1. 1.
    Chris Aholt, Bernd Sturmfels, and Rekha Thomas: A Hilbert scheme in computer vision, Canad. J. Math. 65 (2013) 961–988.Google Scholar
  2. 2.
    Michel Brion: Multiplicity-free subvarieties of flag varieties, in Commutative algebra (Grenoble/Lyon, 2001), 13–23, Contemp. Math. 331, American Mathematical Society, Providence, RI, 2003.Google Scholar
  3. 3.
    William Fulton: Young tableaux, London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge, 1997.Google Scholar
  4. 4.
    Daniel R. Grayson and Michael E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at
  5. 5.
    Charles M. Jessop: A Treatise on the Line Complex, Cambridge University Press, 1903.Google Scholar
  6. 6.
    Kathlén Kohn, Bernt Ivar Utstøl Nødland, and Paolo Tripoli: Secants, bitangents, and their congruences, in Combinatorial Algebraic Geometry, 87–112, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  7. 7.
    Ernst Kummer: Über die algebraischen Strahlensysteme, insbesondere über die der ersten und zweiten Ordnung, Abh. K. Preuss. Akad. Wiss. Berlin (1866) 1–120.Google Scholar
  8. 8.
    Dudley E. Littlewood and Archibald R. Richardson: Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A 233 (1934) 99–141.Google Scholar
  9. 9.
    Ezra Miller and Bernd Sturmfels: Combinatorial Commutative Algebra, Graduate Texts in Mathematics 227, Springer-Verlag, New York, 2004.Google Scholar
  10. 10.
    Evan D. Nash, Ata Firat Pir, Frank Sottile, and Li Ying: The convex hull of two circles in \(\mathbb{R}^{3}\), in Combinatorial Algebraic Geometry, 1–19, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  11. 11.
    Jean Ponce, Bernd Sturmfels, and Matthew Trager: Congruences and concurrent lines in multi-view geometry, Adv. Appl. Math. 88 (2017) 62–91.Google Scholar
  12. 12.
    Hermann Schubert: Anzahl-Bestimmungen für Lineare Räume, Acta Math. 8 (1886) 97–118.Google Scholar
  13. 13.
    Peter Sturm, Srikumar Ramalingam, Jean-Philippe Tardif, Simone Gasparini, and João Barreto: Camera models and fundamental concepts used in geometric computer vision, Foundations and Trends in Computer Graphics and Vision 6 (2011) 1–183.Google Scholar
  14. 14.
    Matthew Trager, Martial Hebert, and Jean Ponce: The joint image handbook, in Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2015.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA

Personalised recommendations