An example of Fourier--Mukai partners of minimal elliptic surfaces

TL;DR

This paper provides an example of two smooth projective elliptic surfaces over the complex numbers that are derived equivalent but not isomorphic, illustrating the distinction between derived category equivalence and geometric isomorphism.

Contribution

It presents the first known example of relatively minimal elliptic surfaces that are Fourier--Mukai partners without being isomorphic.

Findings

Identifies a pair of elliptic surfaces that are D-equivalent but not isomorphic

Clarifies the relationship between derived categories and geometric isomorphism for elliptic surfaces

Contributes to understanding of derived equivalences in algebraic geometry

Abstract

Let $X$ and $Y$ be smooth projective varieties over $\C$. We say that $X$ and $Y$ are \emph{D-equivalent} (or, $X$ is a \emph{Fourier--Mukai partner} of $Y$) if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories. The aim of this short note is to find an example of mutually D-equivalent but not isomorphic relatively minimal elliptic surfaces.