I will explain a tropical approach to Nagata's conjecture and describe new necessary conditions for a tropical point to be a point of multiplicity m.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/4234
[3] http://www.mpim-bonn.mpg.de/node/4763