Enumerative tropical algebraic geometry pdf

Generally speaking, tropical geometry transforms algebraic or symplectic geometric. This expository paper gives an introduction to these new techniques with a special emphasis on the recent applications to problems in enumerative geometry. Tropical curves, enumerative geometry, gromovwitten invariants, toric surfaces. However, enumerative combinatorics is not just about counting. Although its roots can be traced back to work of g. For a complex plane curve c the idea is simply to restrict it to the open subset c2 of the af. Pdf deformation of tropical hirzebruch surfaces and. With classical complex algebraic geometry in mind the most straightforward way to tropical geometry is via socalled amoebas of algebraic varieties. Tropical geometry is algebraic geometry over the semifield of tropical numbers, i.

Yet they carry information about complex and real varieties. This geometry allows one to replace complex toric varieties with the euclidean nspace and holomorphic curves with certain piecewiselinear graphs there. The result is established with the help of the socalled tropical algebraic geometry. The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. With collaborators, he found tropical analogues of theorems about algebraic curves and further developed the enumerative geometry of curves mi06, mz08. Tropical geometry tropical geometry is a rather new. Enumerative tropical algebraic geometry in r2 grigory mikhalkin abstract. Request pdf tropical algebraic geometry tropical geometry is algebraic geometry over the. 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 set t is a semigroup with respect to tropical addition. It turns out that such curves can be counted by means of.

Groves bg84, it emerged as a subject on its own and received its name as recently as 2002. The paper establishes a formula for enumeration of curves of arbitrary. He classified the degeneration of 1cuspidal curves. Enumerative tropical algebraic geometry in r 2 american. 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. Applications of tropical geometry to enumerative geometry. In \real life, when we talk about counting, we imagine lining up a set of objects and counting them o. Tropical algebraic geometry request pdf researchgate. Markwig, and collaborators have transferred much of gromov. The typical question is to nd the number of objects with a given set of properties. The author would like to acknowledge partial support of the nsf. Enumerative tropical algebraic geometry department of mathematics.

Tropical algebraic geometry there are many examples in. The idea to use tropical curves in enumerative questions, and in particular in. A degeneration of a singular curve on a toric surface, called a tropicalization, was constructed by e. Algebraic and geometric methods in enumerative combinatorics.