Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers II
Abstract
Let K be a Henselian discretely valued field with excellent ring of integers 𝒪K and algebraically closed residue field k. Let X→ℙ1K 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 cdc(y) associated to a multiplicity $2$ point y of the branch divisor of the normalization in K(X) of a regular 𝒪K-model 𝒴 of ℙ1K; modulo several key results about 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 cdc(y). In particular, we show cdc(y) is non-negative except under highly restrictive conditions on n and the structure of the branch divisor at y. Moreover, if cdc(y) is negative, we show the spectrum of the complete local ring of any point lying over y under the normalization of 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 cedc(y), which is used in our joint work with Obus and Srinivasan to give a new proof of a result of Kohls.
Related
- Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I
- The Five-Point Case of Mordell's Discriminant Inequality
- Circular Rearrangement Inequality and Optimal Cyclic Birth and Death Chains
- The Strong Naor--Schechtman Convolution Inequality on the Product of Cyclic Groups
- Brauer-Kuroda relations for ramified graph covers
BibTeX
@misc{stewart2026conductordiscriminant,
title = {{Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers II}},
author = {Connor Stewart},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20585},
eprint = {2609.20585},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20585},
}
RIS
TY - UNPB
TI - Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers II
AU - Connor Stewart
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.20585
UR - https://arxiv.org/abs/2609.20585
AB - Let K be a Henselian discretely valued field with excellent ring of integers 𝒪_K and algebraically closed residue field k. Let X→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 cdc(y) associated to a multiplicity $2$ point y of the branch divisor of the normalization in K(X) of a regular 𝒪_K-model 𝒴 of ℙ^1_K; modulo several key results about 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 cdc(y). In particular, we show cdc(y) is non-negative except under highly restrictive conditions on n and the structure of the branch divisor at y. Moreover, if cdc(y) is negative, we show the spectrum of the complete local ring of any point lying over y under the normalization of 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 cedc(y), which is used in our joint work with Obus and Srinivasan to give a new proof of a result of Kohls.
ER -