# Derived equivalences and Kodaira fibers

## TL;DR

This paper investigates the conditions under which different Kodaira curves are derived equivalent, classifies their Fourier-Mukai partners, and studies the properties of their derived categories of singularities.

## Contribution

It provides necessary conditions for derived equivalence of Kodaira curves, classifies Fourier-Mukai partners, and analyzes the derived categories of singularities for non-reduced curves.

## Key findings

- Necessary conditions for derived equivalence of Kodaira curves
- Classification of Fourier-Mukai partners of reduced Kodaira curves
- Idempotent completeness of derived categories of singularities for certain non-reduced curves

## Abstract

We give necessary conditions for two (including non-reduced and multiple) Kodaira curves to be derived equivalent. We classify Fourier-Mukai partners of any reduced Kodaira curve. We prove that the derived category of singularities of any non-reduced and non-multiple Kodaira curve is idempotent complete.