Enumerative tropical algebraic geometry pdf

This expository paper gives an introduction to these new techniques with a special emphasis on the recent applications to problems in enumerative geometry. The idea to use tropical curves in enumerative questions, and in particular in. The set t is a semigroup with respect to tropical addition. A degeneration of a singular curve on a toric surface, called a tropicalization, was constructed by e. For a complex plane curve c the idea is simply to restrict it to the open subset c2 of the af. Applications of tropical geometry to enumerative geometry. Pdf deformation of tropical hirzebruch surfaces and. Tropical geometry is algebraic geometry over the semifield of tropical numbers, i. Deformation of tropical hirzebruch surfaces and enumerative geometry article pdf available in journal of algebraic geometry 254 march 20 with 38 reads how we measure reads.

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. He classified the degeneration of 1cuspidal curves. The typical question is to nd the number of objects with a given set of properties. This geometry allows one to replace complex toric varieties with the euclidean nspace and holomorphic curves with certain piecewiselinear graphs there. Tropical geometry tropical geometry is a rather new. With collaborators, he found tropical analogues of theorems about algebraic curves and further developed the enumerative geometry of curves mi06, mz08. Enumerative tropical algebraic geometry in r2 grigory mikhalkin abstract.

The result is established with the help of the socalled tropical algebraic geometry. Tropical curves, enumerative geometry, gromovwitten invariants, toric surfaces. Markwig, and collaborators have transferred much of gromov. With classical complex algebraic geometry in mind the most straightforward way to tropical geometry is via socalled amoebas of algebraic varieties. Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebrogeometric problems into purely combinatorial ones. It turns out that such curves can be counted by means of certain lattice paths in the newton polygon. Algebraic and geometric methods in enumerative combinatorics. It turns out that such curves can be counted by means of. The enumerative geometry of rational and elliptic tropical. Geometrically, tropical varieties are much simpler than their classical counterparts.

However, enumerative combinatorics is not just about counting. Tropical algebraic geometry there are many examples in. The author would like to acknowledge partial support of the nsf. The paper establishes a formula for enumeration of curves of arbitrary. Enumerative tropical algebraic geometry department of mathematics. In \real life, when we talk about counting, we imagine lining up a set of objects and counting them o. Enumerative tropical algebraic geometry in r 2 american. Tropical algebraic geometry request pdf researchgate.