A origami of genus 2 with a translation

TL;DR

This paper investigates a specific genus 2 Teichmueller curve derived from an origami with a Klein four group symmetry, providing explicit descriptions, genus calculation, and boundary cusp enumeration.

Contribution

It offers an explicit description of the Teichmueller curve in terms of affine plane curves and analyzes its geometric properties, including genus and boundary cusps.

Findings

The Teichmueller curve is a nonsingular affine curve of genus 0.

Explicit affine plane curve descriptions of points on the curve.

Determined the number of cusps in the boundary of the moduli space.

Abstract

We study an example of a Teichmueller curve in the moduli space of algebraic curves of genus 2 coming from an origami S. It is particular in that its points admit the Klein four group as a subgroup of the automorphism group. We give an explicit description of its points in terms of affine plane curves, we show that the Teichmueller curve is a nonsingular, affine curve of genus 0 and we determine the number of cusps in the boundary of the moduli space.