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A study of general Martens-special chains of cycles

Marc Coppens

Abstract

For a general Martens-special chain of cycles $Γ$ of type $k$ we prove that the gonality is equal to $k+2$. Although $\dim (W^1_{k+2} (Γ))=k$ we prove that $w^1_{k+2}(Γ)=0$. We also compute the gonality sequence of $Γ$ and we prove it is divisorial complete. We prove that a general Martens-special discrete chain of cycles $G$ of type $k$ has the same gonality sequence.

A study of general Martens-special chains of cycles

Abstract

For a general Martens-special chain of cycles of type we prove that the gonality is equal to . Although we prove that . We also compute the gonality sequence of and we prove it is divisorial complete. We prove that a general Martens-special discrete chain of cycles of type has the same gonality sequence.

Paper Structure

This paper contains 6 sections, 15 theorems, 3 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Generalities
  3. The gonality of a general Martens-special chain of cycles
  4. Proof of Theorem B
  5. Proof of Theorem A
  6. The case of discrete chains of cycles

Key Result

Lemma 1

Let $D$ be any divisor of degree $d$ on a chain of cycles of genus $g$. Then $D$ is equivalent to a unique divisor of the form $\sum_{i=1}^g <\xi _i>_i +(d-g).w_g$.

Figures (4)

  • Figure 1: a chain of cycles
  • Figure 2: rectangle $[12 \times 3]$
  • Figure 3: A finite cycle of length 5
  • Figure 4: A discrete chain of cycles of genus 3

Theorems & Definitions (53)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Theorem 1
  • Definition 5
  • Definition 6
  • Proposition 1
  • proof : Proof of Proposition \ref{['prop1']}
  • ...and 43 more