This thesis studies three particular types polytopal subdivisions with concrete applica- tions to other mathematical objects, particularly in algebraic geometry.
The first type of polytopal subdivision consists in hypersimplicial subdivisions. These are subdivisions induced by linear projections of hypersimplices. In the case where the projection sends the canonical bases to the vertices of a convex polygon, hypersimplicial subdivisions are in bijection with Grassmannian graphs, a type of planar graph that appears in work of Postnikov to parametrize the positive Grassmannian. We show that for these cases, the poset of hypersimplical subdivions, and hence the poset of Grassmannian graphs of a given type, is homotopic to a sphere, solving a question of Postnikov. For more general projections of hypersimplices, we study the fiber polytope and show that in some cases it is normally equivalent to the Minkowski sum of some the faces of the corresponding secondary polytope.
The second type of polytopal subdivision is regular matroid subdivisions. These objects are polytopal complexes dual to tropical linear spaces. Given any matroid polytope, functions from the set of vertices to the reals that induce a matroid subdivision are the tropical analog of a Plücker vector. We study the Dressian of that matroid, which is the space of all such tropical Plücker vectors. It is a subfan of the secondary fan of the matroid polytope. We show that matroid subdivisions are determined by its 3-skeleton. We study tropical linear spaces arising from matrices with tropical entries, called Stiefel tropical linear spaces. We show that these are a valuated analog of transversal matroids, generalizing much of the theory of transversal matroids to the valuated case. In particular, we concretely describe the space of all tropical matrices with the same tropical Plücer vector. In the process, we show that transversality is a ‘local’ property.
The third type of polytopal subdivisions is regular lattice polygon subdivisions. We make use of them to study Harnack curves. We generalize to arbitrary toric surfaces work of Kenyon and Okounkov, who computed the moduli space of Harnack curves in the projective plane with a given degree. Then we use the fact that Harnack curves can be constructed using regular lattice polygon subdivisions via Viro’s patchworking method to construct a meaningful compactification of the moduli space of Harnack curves. In the process we also make use of abstract tropical curves, which exhibits again the duality between polygon subdivisions and tropical varieties. The result is a compact moduli space of Harnack curves that has a cell complex structure with the same poset as the secondary polytope of the Newton polygon.
