- Series
- Algebra Seminar
- Time
- Monday, October 21, 2013 - 3:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- María Angélica Cueto – Columbia University – http://math.columbia.edu/~macueto/
- Organizer
- Anton Leykin

Fix a complete non-Archimedean valued field K. Any subscheme X of
(K^*)^n can be "tropicalized" by taking the (closure) of the
coordinate-wise valuation. This process is highly sensitive to
coordinate changes. When restricted to group homomorphisms between the
ambient tori, the image changes by the corresponding linear map. This
was the foundational setup of tropical geometry.
In recent years the picture has been completed to a commutative
diagram including the analytification of X in the sense of Berkovich.
The corresponding tropicalization map is continuous and surjective and
is also coordinate-dependent. Work of Payne shows that the Berkovich
space X^an is homeomorphic to the projective limit of all
tropicalizations. A natural question arises: given a concrete X, can
we find a split torus containing it and a continuous section to the
tropicalization map? If the answer is yes, we say that the
tropicalization is faithful.
The curve case was worked out by Baker, Payne and Rabinoff. The
underlying space of an analytic curve can be endowed with a
polyhedral structure locally modeled on an R-tree with a canonical
metric on the complement of its set of leaves. In this case, the
tropicalization map is piecewise linear on the skeleton of the curve
(modeled on a semistable model of the algebraic curve). In higher
dimensions, no such structures are available in general, so the
question of faithful tropicalization becomes more challenging.
In this talk, we show that the tropical projective Grassmannian of
planes is homeomorphic to a closed subset of the analytic Grassmannian
in Berkovich sense. Our proof is constructive and it relies on the
combinatorial description of the tropical Grassmannian (inside the
split torus) as a space of phylogenetic trees by Speyer-Sturmfels. We
also show that both sets have piecewiselinear structures that are
compatible with our homeomorphism and characterize the fibers of the
tropicalization map as affinoid domains with a unique Shilov boundary
point. Time permitted, we will discuss the combinatorics of the
aforementioned space of trees inside tropical projective space.
This is joint work with M. Haebich and A. Werner (arXiv:1309.0450).