Location of the real cubic surfaces in the parameter space

Manuel Falconi, Ernesto A. Lacomba, Jaume Llibre

Research output: Contribution to journalArticleResearchpeer-review


We consider the space C of non-singular cubic surfaces W(a, b, c, d) which are obtained as intersection of the sphere §3 with the algebraic variety x3 + y3 + z3 + w3 + (ax + by + cz+ dw)3 = 0. By using some symmetries, we determine ten different classes of equivalence of C. Through a simple geometrical method, based on the observation that the problem can be reduced to enumerating the connected components of the complement of a certain piecewise-linear set in the 4-dimensional space, we give a partition of R4 in accordance with the different equivalence classes. The novelty of our approach is the non-algebraic way with which the problem is treated. © 2008 Birkhäuser Verlag Basel/Switzerland.
Original languageEnglish
Pages (from-to)147-167
JournalQualitative Theory of Dynamical Systems
Publication statusPublished - 1 Aug 2008


  • Bifurcation
  • Cubic surfaces
  • Non-singular varieties
  • Normal form


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