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Brief introduction to tropical geometry
The paper consists of lecture notes for a mini-course given by the authors at
the G\"okova Geometry \& Topology conference in May 2014. We start the
exposition with tropical curves in the plane and their applications to problems
in classical enumerative geometry, and continue with a look at more general
tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise
A Tropical Toolkit
We give an introduction to Tropical Geometry and prove some results in
Tropical Intersection Theory. The first part of this paper is an introduction
to tropical geometry aimed at researchers in Algebraic Geometry from the point
of view of degenerations of varieties using projective not-necessarily-normal
toric varieties. The second part is a foundational account of tropical
intersection theory with proofs of some new theorems relating it to classical
intersection theory.
Revised version includes many corrections, more examples, and improved
exposition.Comment: 38 page
A bit of tropical geometry
This friendly introduction to tropical geometry is meant to be accessible to
first year students in mathematics. The topics discussed here are basic
tropical algebra, tropical plane curves, some tropical intersections, and
Viro's patchworking. Each definition is explained with concrete examples and
illustrations. To a great exten, this text is an updated of a translation from
a french text by the first author. There is also a newly added section
highlighting new developments and perspectives on tropical geometry. In
addition, the final section provides an extensive list of references on the
subject.Comment: 27 pages, 19 figure
Towards Tropical Psi Classes
To help the interested reader get their initial bearings, I present a survey of prerequisite topics for understanding the budding field of tropical Gromov-Witten theory. These include the language and methods of enumerative geometry, an introduction to tropical geometry and its relation to classical geometry, an exposition of toric varieties and their correspondence to polyhedral fans, an intuitive picture of bundles and Euler classes, and finally an introduction to the moduli spaces of n-pointed stable rational curves and their tropical counterparts
A Journey to Fuzzy Rings
Enumerative geometry is a very old branch of algebraic geometry. In this thesis, we will describe several classical problems in enumerative geometry and their solutions in order to motivate the introduction of tropical geometry. Finally, fuzzy rings, a powerful algebraic framework for tropical and algebraic geometry is introduced
Berkovich skeleta and birational geometry
We give a survey of joint work with Mircea Musta\c{t}\u{a} and Chenyang Xu on
the connections between the geometry of Berkovich spaces over the field of
Laurent series and the birational geometry of one-parameter degenerations of
smooth projective varieties. The central objects in our theory are the weight
function and the essential skeleton of the degeneration. We tried to keep the
text self-contained, so that it can serve as an introduction to Berkovich
geometry for birational geometers.Comment: These are expanded lecture notes of a talk at the Simons Symposium on
Non-Archimedean Geometry and Tropical Geometry (March 31-April 6, 2013). They
have been submitted to the conference proceeding
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