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

TL;DR

This paper provides an explicit method for resolving divisors on models of the projective line over discretely valued fields, using Mac Lane's valuation theory to describe the minimal embedded resolution.

Contribution

It introduces a concrete approach to explicitly compute minimal embedded resolutions of divisors on models of the projective line using Mac Lane's valuations.

Findings

Explicit description of minimal embedded resolution $ ext{div}_0(f)$

Use of Mac Lane's theory for valuation computation

Application to models over discretely valued fields

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 $\mathcal{Y}$.