Stratification of the moduli space of four--gonal curves

TL;DR

This paper studies the geometry of four-gonal curves by embedding their canonical models into a special ruled surface, leading to a stratification of their moduli space based on four new invariants.

Contribution

It introduces a unique conic-ruled surface associated with four-gonal curves and defines four invariants to stratify their moduli space, with computed dimensions.

Findings

Canonical models lie in a unique conic-ruled surface.

Four invariants are introduced to classify four-gonal curves.

Dimensions of the stratified subvarieties are explicitly computed.

Abstract

Let $X$ be a smooth irreducible projective curve of genus $g$ and gonality 4. We show that the canonical model of $X$ is contained in a uniquely defined surface, ruled by conics, whose geometry is deeply related to that of $X$. This surface allows us to define four invariants of $X$ and hence to stratify the moduli space of four-gonal curves by means of closed irreducible subvarieties whose dimensions we compute.