A study of general Martens-special chains of cycles

TL;DR

This paper investigates the gonality of general Martens-special chains of cycles, establishing its value, the dimension of certain linear systems, and the structure of the gonality sequence, revealing new geometric properties.

Contribution

It proves that the gonality of a general Martens-special chain of cycles of type k is k+2 and computes its gonality sequence, showing it is divisorial complete.

Findings

Gonality equals k+2 for chains of type k

Dimension of W^1_{k+2} is k, but w^1_{k+2} is zero

Gonality sequence is divisorial complete

Abstract

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