Models and Integral Differentials of Hyperelliptic Curves

Let $C:y^2=f(x)$ be a hyperelliptic curve of genus $g\ge 1$, defined over a complete discretely valued field $K$, with ring of integers $O_K$. Under certain conditions on $C$, mild when residue characteristic is not $2$, we explicitly construct the minimal regular model with normal crossings $\mathcal{C}\mathrm{/}{O}_K$ of $C$. In the same setting we determine a basis of integral differentials of $C$, that is an $O_K$-basis for the global sections of the relative dualising sheaf ${\omega }_{\mathcal{C}\mathrm{/}{O}_K}$.