The derived category \(\text{D}^{\text{b}}(X)\) of a variety \(X\) encodes fundamental geometric information and an interesting path in order to decode this information is to study the relationship between the number of Fourier-Mukai partners of a variety (that is, the derived equivalent varieties) and of its canonical cover.
This paper addresses this problem for surfaces of Kodaira dimension \(0\). More precisely, the author proves that the canonical covering of an Enriques surface does not admit non-isomorphic Fourier-Mukai partners and also shows that the canonical cover \(A\) of a bielliptic surface can have at most one non-trivial Fourier-Mukai partner, given by \(\widehat{A}\).
These results are used to establish, respectively, that birational Hilbert schemes of points on K3 surfaces that cover Enriques surfaces are isomorphic and that birational generalised Kummer varieties \(K_m(A)\) and \(K_m(B)\) (\(m\geq 2\)) are such that the abelian surfaces \(A\) and \(B\) are either isomorphic or \(B\) is isomorphic to \(\widehat{A}\).
These interesting results are presented in a nicely written and well organized paper which shows that the geometric information imposes restrictions on the number of Fourier-Mukai partners.