Discriminants and Artin conductors

TL;DR

This paper explores the relationship between discriminants, Artin conductors, and singular fiber invariants in degenerations from projective duality, providing formulas and calculations that connect algebraic and topological data.

Contribution

It establishes a discriminant-different formula linking discriminants to Artin conductors and computes these invariants explicitly for planar curves.

Findings

Derived a discriminant-different formula in algebraic number theory context

Calculated different precisely for planar curve discriminants

Connected multiplicities of discriminants to topological and wild ramification invariants

Abstract

We study questions of multiplicities of discriminants for degenerations coming from projective duality over discrete valuation rings. The main result is a type of discriminant-different formula in the sense of classical algebraic number theory, and we relate it to Artin conductors via Bloch's conjecture. In the case of discriminants of planar curves we can calculate the different precisely. In general these multiplicities encode topological invariants of the singular fibers and in the case of characteristic $p$, wild ramification data in the form of Swan conductors.