Determinantal representations of smooth cubic surfaces

Geometriae Dedicata

Volumn 125, Issue 1, 2007, Pages 115-140

  Buckley, Anita ^a   Košir, Tomaž ^a  

^a UNIVERSITY OF LJUBLJANA   (Slovenia)

Author keywords

Determinantal representations; Linear systems of twisted cubic curves; Real smooth cubic surfaces; Schlaefli's double six; Self adjoint and definite determinantal representations; Smooth cubic surface

Indexed keywords

EID: 34547221205     PISSN: 00465755     EISSN: 15729168     Source Type: Journal    
DOI: 10.1007/s10711-007-9144-x     Document Type: Article

References (27)

1. Beauville A. (2000). Determinantal hypersurfaces. Michigan Math. J. 48: 39-63
2. Brundu M. and Logar A. (1998). Parametrization of the orbits of cubic surfaces. Transform Groups 3(3): 209-239
3. Brundu, M., Logar A.: Classification of cubic surfaces with computational methods. Quaderni Matematici, Serie II, No. 375, Dept. of Math., Univ. of Triest (1996)
4. Cayley A. (1849). On the triple tangent planes of surfaces of the third order. Cambridge and Dublin Math J 4: 118-138
5. Clebsch A. (1866). Die Geometrie auf den Flächen dritter Ordnung. J. f. d. Reine Ang Math 65: 359-380
6. Cook R.J. and Thomas A.D. (1979). Line bundles and homogeneous matrices. Quat J Math Oxford Ser (2) 30: 423-429
7. Cremona L. (1868). Mémoire de géométrie pure sur les surfaces du troisiéme ordre. J des Mathématiques pures et appliquées 68: 1-133
8. Dickson L.E. (1921). Determination of all general homogeneous polynomials expressible as determinants with linear elements. Trans. Amer. Math. Soc. 22: 167-179
9. Dixon A.C. (1900-1902). Note on the reduction of ternary quartic to a symmetrical determinant. Proc. Cambridge Phil. Soc. 2: 350-351
10. Dolgachev, I.V.: Luigi Cremona and cubic surfaces. In: Luigi Cremona (1830-1903). Convegno di Studi matematici, Istituto Lombardo, Accademia di Scienze e Lettere, Milano pp. 55-70 (2005)
11. Dolgachev, I.V.: Topics in Classical Algebraic Geometry. Part I, www.math.lsa.umich.edu/idolga/ lecturenotes.html (2006)
12. 3. In The Curves Seminar at Queen's. Vol. VI, Queen's Papers in Pure and Applied Math. No. 83, Queen's University, Kingston, Canada pp. A1-A99 (1989)
13. Harris, J.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 133. Springer-Verlag (1992)
14. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer-Verlag (1977)
15. Henderson A. (1911). The 27 lines upon the Cubic Surface. Hafner Publishing, New York
16. Hilbert, D., Cohn-Vossen, S.: Geometry and Imagination. Chelsea Publishing (1952) (Reprinted in 1999 by AMS)
17. Kollár, J., Smith, K.E., Corti, A.: Rational and Nearly Rational Varieties. Cambridge Studies in Advanced Mathematics, vol. 92. Cambridge University Press (2004)
18. Košir T. (2003). The Cayley-Hamilton theorem and inverse problems for multiparameter systems. Lin. Alg. Appl 367: 155-163
19. Livšic, M.S., Kravitsky, N., Markus, A.S., Vinnikov, V.: Theory of Commuting Nonselfadjoint Operators. Kluwer Academic Publishers (1995)
20. Reid, M.: Undergraduate Algebraic Geometry. London Mathematical Society Student Texts, vol. 12. Cambridge University Press (1988)
21. Room, T. G: The Geometry of Determinantal Loci. Cambridge University Press (1938)
22. Schröter H. (1863). Nachweis der 27 Geraden auf der allgemeinen Oberfläche dritter Ordnung. J. f. d. Reine Ang. Math. 62: 265-280
23. Segre, B.: The Non-Singular Cubic Surfaces. Oxford University Press (1942)
24. Shafarevich, I.R.: Basic Algebraic Geometry 1. Springer-Verlag (1994)
25. Vinnikov V. (1986). Self-adjoint determinantal representations of real irreducible cubics. Oper Theory: Adv. Appl. 19: 415-442
26. Vinnikov V. (1989). Complete description of determinantal representations of smooth irreducible curves. Lin Alg Appl 125: 103-140
27. Vinnikov V. (1993). Self-adjoint determinantal representations of real plane curves. Math. Anal. 296: 453-479