Weierstrass semigroups on double covers of genus four curve

TL;DR

This paper classifies all possible Weierstrass semigroups at ramification points on double covers of a genus 4 curve with genus greater than 11, and constructs examples realizing these semigroups.

Contribution

It provides a complete classification of Weierstrass semigroups for ramification points on certain double covers of genus 4 curves and constructs explicit examples.

Findings

All possible Weierstrass semigroups identified

Explicit constructions of double covers with these semigroups

Results apply to covers with genus > 11

Abstract

Let C be a complete non-singular irreducible curve of genus 4 over an algebraically closed field of characteristic 0. We determine all possible Weierstrass semigroups of ramification points on double covers of C which have genus greater than 11. Moreover, we construct double covers with ramification points whose Weierstrass semigroups are the possible ones.