Fourier-Mukai partners of singular genus one curves

TL;DR

This paper proves that for certain singular genus one curves with trivial dualising sheaf, any Fourier-Mukai partner must be isomorphic to the original curve, extending known results from smooth elliptic curves.

Contribution

The paper extends the classification of Fourier-Mukai partners to singular genus one Gorenstein curves with trivial dualising sheaf, showing they are uniquely determined.

Findings

Fourier-Mukai partners of these curves are isomorphic to the original

The result generalizes known theorems from smooth elliptic curves

Provides new insights into derived equivalences of singular curves

Abstract

The objective of the paper is to prove that, as it happens for smooth elliptic curves, any Fourier-Mukai partner of a projective reduced Gorenstein curve of genus one and trivial dualising sheaf, is isomorphic to itself.