Doctor of Philosophy / Mathematics / Ilia Zharkov / A smooth affine hypersurface of complex dimension n is homotopy equivalent to a real n-dimensional cell complex. We describe a recipe of constructing such cell complex for the hypersurfaces of dimension 1 and 2, i.e. for curves and surfaces. We call such cell complex a skeleton of the hypersurface.
In tropical geometry, to each hypersurface, there is an associated hypersurface, called tropical hypersurface given by degenerating a family of complex amoebas. The tropical hypersurface has a structure of a polyhedral complex and it is a base of a torus fibration of the hypersurface constructed by Mikhalkin. We introduce on the edges of a tropical hypersurface an orientation given by the gradient flow of some piece-wise linear function. With the help of this orientation, we choose some sections and fibers of the fibration.These sections and fibers constitute a cell complex and we prove that this complex is the skeleton by using decomposition of the coemoeba of a classical pair-of-pants. We state and prove our main results for the case of curves and surfaces in Chapters 4 and 5.
| Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/20395 |
| Date | January 1900 |
| Creators | Thapa Magar, Surya |
| Publisher | Kansas State University |
| Source Sets | K-State Research Exchange |
| Detected Language | English |
| Type | Dissertation |
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