  Let $C$ be a smooth projective curve, $E$ a locally free sheaf. Hyperquot
schemes on $C$ parametrise flags of coherent quotients of $E$ with fixed
Hilbert polynomial, and offer alternative compactifications to the spaces of
maps from $C$ to partial flag varieties. Motivated by enumerative geometry, in
this paper we construct a perfect obstruction theory (and hence a virtual class
and a virtual structure sheaf) on these moduli spaces, which we use to provide
criteria for smoothness and unobstructedness. Under these assumptions, we
determine their motivic partition function in the Grothendieck ring of
varieties, in terms of the motivic zeta function of $C$.
