  Fix a ruled surface S obtained as the projective completion of a line bundle
L on a complex elliptic curve; we study the moduli problem of parametrizing
certain pairs consisting of a sheaf E on S and a map of E to a fixed reference
sheaf on S. We prove that the full moduli stack for this problem is
representable by a scheme in some cases. Moreover, the moduli stack admits an
action by the group ${\mathbf C}^*$, and we determine its fixed-point set,
which leads in some special cases to explicit formulas for the rational
homology of the moduli space.
