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DEPARTMENTAL PHD THESIS EXAM

Wednesday, April 25, 2012, 2:10 p.m., in BA 6183, 40 St. George St.

PhD Candidate:  Jihyeon Jessie Yang

PhD Advisor:  Askold Khovanskii

Thesis Title:  Tropical Severi Varieties and Applications
(http://www.math.toronto.edu/jyang/PhDThesisJYANG.pdf)

Thesis Abstract:

The main topic of this thesis is the tropicalizations of Severi varieties, 
which we call tropical Severi varieties. Severi varieties are classical 
objects in algebraic geometry. They are parameter spaces of plane nodal 
curves. On the other hand, tropicalization is an operation defined in 
tropical geometry, which turns subvarieties of an algebraic torus
into certain polyhedral objects in real vector spaces. By studying 
the tropicalizations, it may be possible to transform algebro-geometric 
problems into purely combinatorial ones.  Thus, it is a natural question, 
"what are tropical Severi varieties?" In this thesis, we give
a partial answer to this question: we obtain a description of tropical 
Severi varieties in terms of regular subdivisions of polygons. Given 
a regular subdivision of a convex lattice polygon, we construct an 
explicit parameter space of plane curves. This parameter space
is much simpler object than the corresponding Severi variety and it is 
closely related to a flat degeneration of the Severi variety, which in 
turn describes the tropical Severi variety.

We present two applications. First, we understand G. Mikhalkin's 
correspondence theorem for the degrees of Severi varieties in terms 
of tropical intersection theory. In particular, this provides a proof 
of the independence of point-configurations in the enumeration
of tropical nodal curves. The second application is about Secondary fans. 
Secondary fans are purely combinatorial objects which parametrize all 
the regular subdivisions of polygons. We provide a relation between 
tropical Severi varieties and Secondary fans.
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