On the refinements of a polyhedral subdivision

• Autores: Francisco de Santos
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 52, Fasc. 3, 2001, págs. 231-256
• Idioma: inglés
• Títulos paralelos:
  • Refinamientos de una subdivisión poliedral
• Enlaces
  • Texto completo
• Resumen

  • Let pi: P --> Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by pi (or pi-induced) and the fiber polytope of the projection: a polytope Sygma(P,pi) of dimension dim(P)-dim(Q) whose faces are in correspondence with the coherent pi-induced subdivisions (or pi-coherent subdivisions). In this paper we investigate the structure of the poset of pi-induced refinements of a pi-induced subdivision. In particular, we define the refinement polytope associated to any pi-induced subdivision S, which is a generalization of the fiber polytope and shares most of its properties. As applications to the theory we prove that if a point configuration has non-regular subdivisions then it has non-regular triangulations and we provide simple proofs of the existence of non-regular subdivisions for many particular point configurations.

• Referencias bibliográficas
  • V. Alexeev, Complete moduli in the presence of semiabelian group action, preprint 1999, to appear in J. Algebraic Geom. Available at Los Alamos...
  • C.A. Athanasiadis, J.A. de Loera, V. Reiner, and F. Santos, Fiber polytopes for the projections between cyclic polytopes, in “Combinatorics...
  • C.A. Athanasiadis, J. Rambau, and F. Santos, The generalized Baues problem for cyclic polytopes II, in “Geometric combinatorics (Kotor, 1998)”,...
  • M. Azola and F. Santos, The Baues conjecture in corank 3, Topology, to appear.
  • M. Azola and F. Santos, The graph of triangulations of a point configuration with $d+4$ vertices is 3-connected, Discrete Comput. Geom....
  • L.J. Billera and B. Sturmfels, Fiber polytopes, Ann. of Math. (2) 135 (1992), 527–549.
  • L.J. Billera, M.M. Kapranov, and B. Sturmfels, Cellular strings on polytopes, Proc. Amer. Math. Soc. 122 (1994), 549–555.
  • L.J. Billera, P. Filliman, and B. Sturmfels, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990), 155–179.
  • A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G.M. Ziegler, Oriented Matroids, Cambridge University Press, Cambridge, 1993.
  • J.A. de Loera, Nonregular triangulations of products of simplices, Discrete Comput. Geom. 15 (1996), 253–264.
  • J.A. de Loera, S. Hosten, F. Santos, and B. Sturmfels, The polytope of all triangulations of a point configuration, Doc. Math. J. 1 (1996),...
  • J.A. de Loera, F. Santos, and J. Urrutia, The number of geometric bistellar neighbors of a triangulation, Discrete Comput. Geom. 21 (1999),...
  • P. Edelman and V. Reiner, Visibility complexes and the Baues problem for triangulations in the plane, Discrete Comput. Geom. 20 (1998), 35–59.
  • I.M. Gel'fand, M.M. Kapranov, and A.V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser Boston, Inc.,...
  • B. Huber, J. Rambau, and F. Santos, The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings, J. Eur. Math....
  • M.M. Kapranov, B. Sturmfels, and A.V. Zelevinsky, Quotients of toric varieties, Math. Ann. 290 (1991), 643–655.
  • C.W. Lee, Regular triangulations of convex polytopes, in Applied geometry and discrete mathematics, Amer. Math. Soc., Providence, R.I., (1991),...
  • W.S. Massey, A basic course in algebraic topology, Springer-Verlag, New York, 1991.
  • J. Rambau, Triangulations of cyclic polytopes and higher Bruhat orders, Mathematika 44 (1997), 162–194.
  • J. Rambau and F. Santos, The generalized Baues problem for cyclic polytopes I, in Combinatorics of polytopes, European J. Combin. 21 (2000),...
  • J. Rambau and G.M. Ziegler, Projections of polytopes and the generalized Baues conjecture, Discrete Comput. Geom. 16 (1996), 215–237.
  • V. Reiner, The generalized Baues problem, in New perspectives in algebraic combinatorics, Math. Sci. Res. Inst. Publ. 38 (1999), 293–336.
  • F. Santos, Triangulations of oriented matroids, Mem. Amer. Math. Soc., in press.
  • F. Santos, A point set whose space of triangulations is disconnected, J. Amer. Math. Soc. 13 (2000), 611–637.
  • B. Sturmfels, On the Newton polytope of the resultant, J. Algebraic Combin. 3 (1994), 207–236.
  • G.M. Ziegler, Lectures on polytopes, Springer-Verlag, New York, 1995.