Tropical geometry in enumerative geometry

From CRCG-Wiki

Tropical geometry in enumerative geometry

In enumerative geometry, we fix a geometric object and some conditions that we require our object to satisfy. The conditions have to be chosen in such a way that indeed only a finite number of objects satisfy them. Then we count how many of our objects satisfy the conditions. A well-known problem of enumerative geometry is to determine the numbers N(d,g) of genus g plane curves of degree d passing through 3d+g-1 points in general position. (These numbers do not depend on the position of the 3d+g-1 points, as long as it is sufficiently general.) The numbers N(d,g) are known as Gromov-Witten invariants of the projective plane and were first computed by Maxim Kontsevich for rational curves (i.e. N(d,0)). The same question can be posed tropically. Grigory Mikhalkin's famous Correspondence Theorem states that the tropical numbers equal their algebaic counterparts (see here). This result leads to a new approach for enumerative geometry, counting tropical curves instead of algebraic curves.

This picture shows how the number N(3,0)=12 can be computed by counting tropical curves.

Interestingly, not only the numbers, but also many techniques to determine the numbers carry over to the tropical world. For example, Andreas Gathmann and Hannah Markwig gave tropical proofs of algorithms to determine such numbers (see here or here). The tropical methods developed by Andreas Gathmann and Hannah Markwig were used by Ilya Itenberg, Viatcheslav Kharlamov and Eugenii Shustin to produce a fast tropical algorithm for computing Welschinger invariants (see here).