Explicit minimal embedded resolutions of divisors on models of the projective line

Abstract

Let $K$ be a discretely valued field with ring of integers $\mathcal{O}_K$ with perfect residue field. Let $K(x)$ be the rational function field in one variable. Let $\mathbb{P}^1_{\mathcal{O}_K}$ be the standard smooth model of $\mathbb{P}^1_K$ with coordinate $x$ on irreducible special fiber. Let $f(x) \in \mathcal{O}_K[x]$ be a monic irreducible polynomial with corresponding divisor of zeroes $\text{div}_0(f)$ on $\mathbb{P}^1_{\mathcal{O}_K}$. We give an explicit description of the minimal embedded resolution $\mathcal{Y}$ of the pair $(\mathbb{P}^1_{\mathcal{O}_K}, \text{div}_0(f))$ by using Mac Lane's theory to write down the discrete valuations on $K(x)$ corresponding to the irreducible components of the special fiber of $\mathcal{Y}$.