Relating Different Definitions of Linear Series on Tropical Curves

Abstract

We investigate relationships among several recent notions of linear series on tropical curves, including tropical linear series, strongly recursive tropical linear series of Farkas, Jensen, and Payne, and combinatorial limit linear series of Amini and Gierczak. We introduce structured and locally weakly recursive tropical linear series, showing that every locally weakly recursive tropical linear series is a combinatorial limit linear series. Consequently, every strongly recursive tropical linear series is a combinatorial limit linear series. We also obtain an equivalent characterization of combinatorial limit linear series. We establish additional extensions of these results and construct counterexamples showing that the converse implications fail for strongly recursive tropical linear series. Finally, we investigate the extent to which permutation arrays can arise as local combinatorial data of tropical linear series.