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Obstructions to Relative Tropical Lifting

Algebra & Discrete Mathematics

Speaker: Tristram Bogart, SFSU
Location: 2212 MSB
Start time: Thu, Apr 14 2011, 4:10PM

Tropical geometry is an emerging approach to algebraic geometry that uses combinatorial and discrete geometric techniques. For example, algebraic curves are modelled by certain types of (embedded) graphs and surfaces by certain complexes of polygons. These discrete objects are known as tropical varieties. There is a tropicalization map that takes an ordinary algebraic variety X to a tropical variety trop(X). However, not every tropical variety arises this way, so the tropical lifting problem is to determine which tropical varieties arise as tropicalizations. In general this problem is very difficult but is understood in special cases such as linear spaces and curves of small genus.

In joint work with Eric Katz, we produce an obstruction for certain cases of the related problem of relative tropical lifting: given two tropical varieties S contained in T, both of which arise individually as tropicalizations, can we find varieties X contained in Y so that trop(X) = S and trop(Y) = T? Our obstruction explains a mysterious class of examples, discovered by Vigeland, of tropical surfaces that contain "too many" tropical lines.