Models and integral differentials of hyperelliptic curves

Let $C\; : \;y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 1$ , 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 $\mathcal{C}/O_K$ 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 $\omega _{\mathcal{C}/O_K}$ .