The weighted moduli spaces of sextics

Abstract

We use the weighted moduli height as defined in \cite{sh-h} to investigate the distribution of fine moduli points in the moduli space of genus two curves. We show that for any genus two curve with equation $y^2=f(x)$, its weighted moduli height $\mathfrak{h}(\mathfrak{p})\le 2^3\sqrt{3\cdot 5\cdot 7}\cdot H(f)$, where $H(f)$ is the minimal naive height of the curve as defined in \cite{height}. Based on the weighted moduli height $\mathfrak{h}$ we create a database of genus two curves defined over $\mathbb{Q}$ with small $\mathfrak{h}$ and show that for small such height ($\mathfrak{h}<5$) about 30% of points are fine moduli points.