Thetanulls of cyclic curves of small genus

TL;DR

This paper investigates relations among thetanulls of cyclic curves with specific automorphism properties, classifies genus 2 and 3 curves within certain moduli spaces, and advances understanding of their algebraic and geometric structures.

Contribution

It classifies genus 2 and 3 cyclic curves with automorphisms by deriving relations among their thetanulls, extending classical and recent work on cyclic curves.

Findings

Classification of genus 2 and 3 curves in specific moduli spaces

Explicit relations among thetanulls for these curves

Identification of all possible signatures and automorphism groups

Abstract

We study relations among the classical thetanulls of cyclic curves, namely curves $\mathcal X$ (of genus $g(\mathcal X)>1$) with an automorphism $\sigma$ such that $\sigma$ generates a normal subgroup of the group $G$ of automorphisms, and $g (\mathcal X/ < \sigma>) =0$. Relations between thetanulls and branch points of the projection are the object of much classical work, especially for hyperelliptic curves, and of recent work, in the cyclic case. We determine the curves of genus 2 and 3 in the locus $\mathcal M_g (G, \textbf{C})$ for all $G$ that have a normal subgroup $<\s>$ as above, and all possible signatures \textbf{C}, via relations among their thetanulls.