Advertisement

Mathematical Programming

, Volume 151, Issue 2, pp 585–612 | Cite as

Quartic spectrahedra

  • John Christian Ottem
  • Kristian Ranestad
  • Bernd Sturmfels
  • Cynthia VinzantEmail author
Full Length Paper Series B

Abstract

Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24. This set is stratified by the location of the ten nodes of the corresponding real quartic surface. There are twenty maximal strata, identified recently by Degtyarev and Itenberg, via the global Torelli Theorem for real K3 surfaces. We here give a new proof that is self-contained and algorithmic. This involves extending Cayley’s characterization of quartic symmetroids, by the property that the branch locus of the projection from a node consists of two cubic curves. This paper represents a first step towards the classification of all spectrahedra of a given degree and dimension.

Mathematics Subject Classification

14P10 14P25 52B55 90C22 

Notes

Acknowledgments

Bernd Sturmfels was supported by the NSF (DMS-0968882) and the Max-Planck Institute für Mathematik in Bonn. Cynthia Vinzant was supported by an NSF postdoc (DMS-1204447).

References

  1. 1.
    Blekherman, G., Hauenstein, J., Ottem, J.C., Ranestad, K., Sturmfels, B.: Algebraic boundaries of Hilbert’s SOS cones. Compos. Math. 148, 1717–1735 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Blekherman, G., Parrilo, P., Thomas, R.: Semidefinite Optimization and Convex Algebraic Geometry, MOS-SIAM Series on Optimization 13. SIAM, Philadelphia (2013)Google Scholar
  3. 3.
    Cayley, A.: A memoir on quartic surfaces. In: Proceedings of the London Mathematical Society, vol. 3 (1869/71), pp. 19–69. [Collected Papers, VII, 133–181; see also the sequels on pages 256–260, 264–297]Google Scholar
  4. 4.
    Coble, A.: Algebraic Geometry and Theta Functions. American Mathematical Society, vol. X. Colloquium Publications, New York (1929)Google Scholar
  5. 5.
    Cossec, F.R.: Reye congruences. Trans. Am. Math. Soc. 280, 737–751 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Degtyarev, A., Itenberg, I.: On real determinantal quartics. In: Proceedings of the Gökova Geometry-Topology Conference 2010, pp. 110–128. Int. Press, Somerville, MA (2011)Google Scholar
  7. 7.
    Dixon, A.C.: Note on the reduction of a ternary quartic to a symmetrical determinant. Math. Proc. Camb. Philos. Soc. 11, 350–351 (1902)zbMATHGoogle Scholar
  8. 8.
    Dolgachev, I.V.: Classical Algebraic Geometry. A Modern View. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Friedland, S., Robbin, J.W., Sylvester, J.H.: On the crossing rule. Commun. Pure Appl. Math. 37, 19–37 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gårding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Harris, J., Tu, L.: On symmetric and skew-symmetric determinantal varieties. Topology 23, 71–84 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  13. 13.
    Hudson, R.W.H.T.: Kummer’s Quartic Surface. Cambridge University Press, Cambridge (1905). Reprinted in Cambridge Mathematical Library (1990)zbMATHGoogle Scholar
  14. 14.
    Huh, J. : A counterexample to the geometric Chevalley-Warning conjecture, arXiv:1307.7765.
  15. 15.
    Jessop, G.M.: Quartic Surfaces with Singular Points. Cambridge University Press, Cambridge (1916)zbMATHGoogle Scholar
  16. 16.
    Plaumann, D., Sturmfels, B., Vinzant, C.: Quartic curves and their bitangents. J. Symb. Comput. 46, 712–733 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Plaumann, D., Vinzant, C.: Determinantal representations of hyperbolic plane curves: an elementary approach. J. Symb. Comput. 57, 48–60 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Renegar, J.: Hyperbolic programs and their derivative relaxations. Found. Comput. Math. 6, 59–79 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Sanyal, R.: On the derivative cones of polyhedral cones. Adv. Geom. 13, 315–321 (2013)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • John Christian Ottem
    • 1
  • Kristian Ranestad
    • 2
  • Bernd Sturmfels
    • 3
  • Cynthia Vinzant
    • 4
    Email author
  1. 1.University of CambridgeCambridgeUK
  2. 2.University of OsloOsloNorway
  3. 3.University of CaliforniaBerkeleyUSA
  4. 4.North Carolina State UniversityRaleighUSA

Personalised recommendations