Linear systems on tropical curves

By Christian Haase, Gregg Musiker, Josephine Yu. Mathematische Zeitschrift, 2009

92 citations recorded by the source corpus.

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 |D| of a divisor D on a tropical curve Γ analogously to the classical counterpart. We investigate the structure of |D| 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, |D| 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 |D| is base point free. The tropical convex hull of the image realizes the linear system |D| 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}$$\end{document} -tropical curve is a direct limit of critical groups of finite graphs converging to the curve.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7f9914c8fd84716f6bed7c545ab472e2f6bb7c71

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Citations in the corpus

Degeneration of Linear Series from the Tropical Point of View and Applications (2015, background): “See [HMY12] for a detailed discussion of the tropical semimodule structure on R(D).”
Tropical independence II: The maximal rank conjecture for quadrics (2015, background): “The tropical linear series R ( D ) is a tropical module, which means that it is closed under scalar addition and pointwise minimum [Haase et al. 2012, Lemma 4].”
Metric properties of the tropical Abel–Jacobi map (2009, methodology): “ect limit is an exact functor, the vertical arrows give an injection lim ! J(G) ,!lim ! (G) =H 1(G;Z); (2.6) 2A special case of Theorem 2.10 has been proved independently by Haase, Musiker, and Yu in [10]. 3In [5], this is called a walk. There it is called a path if e j 1 6= e j for j= 2;:::;n. 10 w”
A tropical proof of the Brill-Noether Theorem (2010, background): “The set of effective divisors on Γ that are equivalent to D is naturally identified with the underlying set of a finite polyhedral complex [15].”
Linear series on metrized complexes of algebraic curves (2012, background): “uch that DE +div(f) 0, where E= (s+1)(v) E v. This shows that D (s+1)(v)+div(f ) 0. For a generic choice of E v, we can assume that the slopes of f on the edges adjacent to v are xed [GK] (see also [HMY] and [A, Section 2.1] for a discussion of the cell decomposition of jDjfor D2Div()). For the divi”