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Weierstrass gap sequences and their weights on tropical curves

Omid Amini, Shu Kawaguchi

Abstract

Given a divisor on a tropical curve, we associate to each point of the curve a Weierstrass gap sequence. We investigate structural properties of these gap sequences and explore their relationship with the Weierstrass gap sequences of line bundles on algebraic curves via the tropicalization process.

Weierstrass gap sequences and their weights on tropical curves

Abstract

Given a divisor on a tropical curve, we associate to each point of the curve a Weierstrass gap sequence. We investigate structural properties of these gap sequences and explore their relationship with the Weierstrass gap sequences of line bundles on algebraic curves via the tropicalization process.
Paper Structure (23 sections, 31 theorems, 81 equations, 10 figures, 2 tables)

This paper contains 23 sections, 31 theorems, 81 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Divisors on tropical curves
  4. Rank on hyperelliptic tropical curves
  5. Total weight of the Weierstrass loci
  6. Baker's specialization lemma and specialization of Weierstrass points
  7. Effective locus in Pic(X)
  8. D-Weierstrass gap sequences and coordinatewise partial order
  9. D-Weierstrass points and D-Weierstrass gap sequences
  10. Coordinatewise partial order
  11. Proofs of Theorem \ref{['thm:Wloc:finite']}, Corollary \ref{['cor:Wloc:finite']}, and Proposition \ref{['prop:contrast']}
  12. Proof of Theorem \ref{['thm:Wloc:finite']}
  13. Proof of Corollary \ref{['cor:Wloc:finite']}
  14. Proof of Proposition \ref{['prop:contrast']}
  15. From algebraic curves to tropical curves
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a tropical curve of genus $g \geq 2$, and let $D$ be a divisor on $\Gamma$ of degree $d$ and nonnegative rank $r$. Then, for an isolated point $p$ in $\operatorname{WL}(D)$, the Weierstrass gap sequence at $p$ with respect to $D$ is given by In particular, the $D$-Weierstrass weight of $p$ is $D_p(p)-r$. Furthermore, if $\operatorname{WL}(D)$ is finite, then

Figures (10)

  • Figure 1: A dipole graph $B_g$ of genus $g$ (left) and a wheel graph of genus $g$ (right).
  • Figure 2: Chain of three circles.
  • Figure 3: Non-hyperelliptic curves of genus $3$ without cut vertices.
  • Figure 4: Hyperelliptic curve of genus $3$ without cut vertices.
  • Figure 5: Non-hyperelliptic curve of genus $3$ with cut vertices
  • ...and 5 more figures

Theorems & Definitions (41)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 2.1: Tropical Riemann--Roch formula BNGKMZ
  • ...and 31 more