Some more geometry in the secondary fan

  A polyhedral subdivision of finitely many points A in R^d 
is a decomposition of its convex hull into non-overlapping 
polytopes with vertices in A.  It is regular if the polytopes 
are projections of facets of a polyhedron in R^{d+1}.  Gelfand,
Kapranov, and Zelevinsky introduced the secondary fan of A, 
which encodes all regular subdivisions of A.  When A is integral,
Kapranov, Sturmfels, and Zelevinsky showed how this fan encodes
all possible limiting positions of a projective toric variety 
associated to A, under translation by the ambient torus.

  I will talk on work with Postinghel and Villamizar that
generalizes this to the situation where the points of A
are not necessarily integral.  Here, the toric variety is 
replaced by a curvilinear copy of the convex hull in the
simplex.   In this case, one does not have the tools from 
algebraic geometry (Hilbert schemes), and we instead develop
tools based on the geometry of sequences of points in the
secondary fan.  This work was motivated by questions in 
geometric modeling and algebraic statistics which I plan
to mention.