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. Let \(f(x) \in \mathcal {O}_K[x]\) be a squarefree polynomial with corresponding divisor of zeroes \({{\,\mathrm{div}\,}}_0(f)\) on \({\mathbb {P}}^1_{\mathcal {O}_K}\). We give an explicit description of the minimal embedded resolution \(\mathcal {Y}\) of the pair \(({\mathbb {P}}^1_{\mathcal {O}_K}, {{\,\mathrm{div}\,}}_0(f))\) 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 \(\mathcal {Y}\).