  Under some assumptions, we compute the Picard group of moduli of stable
sheaves on Abelian surfaces. Considering relative moduli spaces, it is
sufficient to consider the moduli of stable sheaves on the product of elliptic
curves. By using elementary transformatios (which was used by Friedman to treat
rank 2 moduli spaces on elliptic surfaces), we treat this case. We also show
that some moduli spaces on P^2 are rational.
