Minimal regular models of quadratic twists of genus two curves

Abstract

Let $K$ be a complete discrete valuation field with ring of integers $R$ and residue field $k$ of characteristic $p>2$. We assume moreover that $k$ is algebraically closed. Let $C$ be a smooth projective geometrically connected curve of genus $2$. If $K(\sqrt{D})/K$ is a quadratic field extension of $K$ with associated character $\chi$, then $C^{\chi}$ will denote the quadratic twist of $C$ by $\chi$. Given the minimal regular model $\mathcal X$ of $C$ over $R$, we determine the minimal regular model of the quadratic twist $C^{\chi}$. This is accomplished by obtaining the stable model $\mathcal{C}^{\chi}$ of $C^{\chi}$ from the stable model $\mathcal{C}$ of $C$ via analyzing the Igusa and affine invariants of the curves $C$ and $C^{\chi}$, and calculating the degrees of singularity of the singular points of $\mathcal{C}^{\chi}$.