2015/06/24 by Kristin Shaw, Shaw, Kristin
Computer Science · Mathematics · #Polynomial and algebraic computation #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.1506.07407
We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and tropical Chern cycles and numbers. We provide a new method for constructing tropical surfaces, called the tropical sum, similar to the fiber sum of usual manifolds. We prove a tropical adjunction formula for curves in compact tropical surfaces satisfying a local condition, a partial Castelnuovo-Enriques criterion for contracting (-1)-curves, and also invariance of (p, q)-homology and Chow groups under tropical modification. Finally we prove a tropical version of Noether's formula for compact surfaces constructed from tropical toric surfaces by way of summations and tropical modifications.