Linear systems on tropical curves

A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system {pipe}D{pipe} of a divisor D on a tropical curve Γ analogously to the classical counterpart. We investigate the structure of {pipe}D{pipe} as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, {pipe}D{pipe} defines a map from Γ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to deg(D) when {pipe}D{pipe} is base point free. The tropical convex hull of the image realizes the linear system {pipe}D{pipe} as a polyhedral complex. We show that curves for which the canonical divisor is not very ample are hyperelliptic. We also show that the Picard group of a ℚ -tropical curve is a direct limit of critical groups of finite graphs converging to the curve.