Reduced divisors and embeddings of tropical curves

Abstract

Given a divisor D D on a tropical curve Γ \Gamma , we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system | D | |D| . This is done by providing an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduced-divisor map defines an embedding of the curve into the linear system and, in this way, classify all the tropical curves with a very ample canonical divisor. As an application of the reduced-divisor map, we show the existence of Weierstrass points on tropical curves of genus at least two and present a simpler proof of a theorem of Luo on rank-determining sets of points. We also discuss the classical analogue of the (tropical) reduced-divisor map: For a smooth projective curve C C and a divisor D D of non-negative rank on C C , reduced divisors equivalent to D D define a morphism from C C to the complete linear system | D | |D| , which is described in terms of Wronskians.