A note on local trigonal fibrations

TL;DR

This paper demonstrates that certain hyperelliptic curves can serve as central fibers in degenerations of trigonal curves, revealing a new connection between these classes of algebraic curves.

Contribution

It establishes that any hyperelliptic curve can be realized as a central fiber in a degeneration of trigonal curves, expanding understanding of their geometric relationships.

Findings

Hyperelliptic curves can be central fibers in trigonal degenerations

Some central fibers of hyperelliptic degenerations are trigonal curves

Any hyperelliptic curve can appear as a degeneration central fiber

Abstract

We show that some of centeral fibers of degenerations of hyperelliptic curves are realized as those trigonal curves. In particular, any hyperelliptic curve can be the central fiber of a degeneration of trigonal curves.