# The trigonal construction in the ramified case

## TL;DR

This paper explores a geometric construction linking ramified double covers of trigonal curves to étale double covers of tetragonal curves, revealing an isomorphism between their Prym varieties.

## Contribution

It establishes a canonical isomorphism between Prym varieties associated with ramified double covers of trigonal curves and étale double covers of tetragonal curves.

## Key findings

- Prym varieties are canonically isomorphic in the described setting.
- The construction relates ramified and étale double covers via Prym varieties.
- Provides new insights into the geometry of trigonal and tetragonal curves.

## Abstract

To every double cover ramified in two points of a general trigonal curve of genus g, one can associate an \'etale double cover of a tetragonal curve of genus g+1. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties.