Relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano fibrations

Abstract

Abstract. We study the group of relative Fourier-Mukai transforms for Weierstraß fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstraß and Fano or anti-Fano fibrations we describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier-Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier-Mukai transforms and we prove that the number of relative Fourier-Mukai partners of a given abelian scheme over a normal base is finite.