and stable loci for the $\theta$-linearization coincide, $W^{ss}=W^s$, and that ${{\bf G}}$ acts freely on the stable locus. Further, we also assume that $W$ has at worst lci singularities and that $W^s$ is smooth. Hence ${W/\!\!/{{\bf G}}}=[W^s/{{\bf G}}]$ is a nonsingular open substack in the quotient stack $[W/{{\bf G}}]$.

$\ke$-stable quasimaps {#e-qmaps}
----------------------

Let $C$ be a connected, at worst nodal, algebraic curve. A map $f:C\lra[W/{{\bf G}}]$ has a (homology) class $$\beta\in ({{\mathrm{Pic}}}([W/{{\bf G}}]))^\vee=\Hom_\ZZ({{\mathrm{Pic}}}([W/{{\bf G}}]),\ZZ)=\Hom_\ZZ({{\mathrm{Pic}}}^{{\bf G}}(W),\ZZ)$$ given by $\beta(L)=\deg f^*L$. The map $f$ is a [*quasimap to*]{} ${W/\!\!/{{\bf G}}}$ if it sends the generic point of each irreducible component of $C$ to ${W/\!\!/{{\bf G}}}$. We may view the quasimap as a [*rational*]{} map from $C$ to ${W/\!\!/{{\bf G}}}$. The (finitely many) points of $C$ which are sent by $f$ to the complement of ${W/\!\!/{{\bf G}}}$ in $[W/{{\bf G}}]$ are called the [*base-points*]{} of the quasimap.

A class $\beta\in ({{\mathrm{Pic}}}([W/{{\bf G}}]))^\vee$ is said to be $\theta$-[*effective*]{} if it is represented by a quasimap to ${W/\!\!/{{\bf G}}}$. For the purposes of this definition, the domain curve is allowed to have finitely many connected components. The $\theta$-[effective]{} classes form a semigroup, denoted ${\mathrm {Eff}}(W,{{\bf G}},\theta)$. By the boundedness results in [@CKM], for each 