Fourier-Mukai transforms for coherent systems on elliptic curves

We determine all the Fourier–Mukai transforms for coherentsystems consisting of a vector bundle over an elliptic curveand a subspace of its global sections, showing that these transformsare indexed by positive integers. We prove that the naturalstability condition for coherent systems, which depends on aparameter, is preserved by these transforms for small and largevalues of the parameter. By means of the Fourier–Mukaitransforms we prove that certain moduli spaces of coherent systemscorresponding to small and large values of the parameter areisomorphic. Using these results we draw some conclusions aboutthe possible birational type of the moduli spaces. We provethat for a given degree d of the vector bundle and a given dimensionof the subspace of its global sections there are at most d differentpossible birational types for the moduli spaces.