Torsion classes of coherent sheaves on an elliptic or product threefold

Let X be an elliptic threefold admitting a Weierstrass elliptic fibration. We extend the main results of Angeles, Lo, and Van Der Linden in [1] by providing explicit properties characterizing the coherent sheaves contained in the torsion classes constructed there. We utilize the relative Fourier-Mukai transform Φ on the derived category of coherent sheaves to describe the cohomology objects of the transforms of coherent sheaves, and show that this, along with information on the dimension of coherent sheaves, is sufficient information to completely describe the main torsion classes. Our method of proof to show these characterizations is then generalized to produce more examples of torsion classes, and gives another method to produce torsion classes in the category of coherent sheaves.