Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers II

Abstract

Let $K$ be a Henselian discretely valued field with excellent ring of integers $\mathcal{O}_K$ and algebraically closed residue field $k$. Let $X\to\mathbb{P}^1_K$ be a cyclic cover of degree $n$ prime to the characteristic of $k$. In joint work with Obus and Srinivasan (arXiv:2609.20553), we define an integer called the conductor-discriminant contribution $\text{cdc}(y)$ associated to a multiplicity $2$ point $y$ of the branch divisor of the normalization in $K(X)$ of a regular $\mathcal{O}_K$-model $\mathcal{Y}$ of $\mathbb{P}^1_{K}$; modulo several key results about $\text{cdc}(y)$, we prove a conductor-discriminant inequality for $X$, extending previous work of Ogg, Saito, Liu, Srinivasan, and Obus$-$Srinivasan. In this companion paper, we supply the necessary technical results for $\text{cdc}(y)$. In particular, we show $\text{cdc}(y)$ is non-negative except under highly restrictive conditions on $n$ and the structure of the branch divisor at $y$. Moreover, if $\text{cdc}(y)$ is negative, we show the spectrum of the complete local ring of any point lying over $y$ under the normalization of $\mathcal{Y}$ in $K(X)$ is a rational double point. Along the way, we show the non-negativity of a related quantity, the conductor exponent-discriminant contribution $\text{cedc}(y)$, which is used in our joint work with Obus and Srinivasan to give a new proof of a result of Kohls.