$d>0$ the set $$\{\beta\in {\mathrm {Eff}}(W,{{\bf G}},\theta)\; |\: \beta(L_\theta)\leq d\}$$ is finite, see Remark 3.2.10 in loc.cit.

For each positive $\ke\in\QQ$ there is a stability condition on quasimaps; in addition, there is an asymptotic stability condition, denoted $\ke=0+$, in which $\ke$ is allowed to be arbitrarily small (but still positive), see [@CKM], also [@CK0], §2.4.

From now on we assume that $(g,k)\neq(0,0)$. For $\ke\geq 0+$, let ${\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}$ denote the moduli space of $\ke$-stable quasimaps of class $\beta$ from $k$-pointed genus $g$ nodal curves to ${W/\!\!/{{\bf G}}}$. It is shown in [@CKM] that when the triple $(W,{{\bf G}},\theta)$ satisfies the assumptions of §\[quotients\] above, these moduli spaces are Deligne-Mumford stacks, proper over the affine quotient $W/_{\mathrm {aff}}{{\bf G}}$, and carrying canonical perfect obstruction theories. The lci condition on $W$ is necessary for the perfectness of the obstruction theory when $\ke\leq 1$. The space ${\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}$ is potentially nonempty only when $$2g-2+k+{{\varepsilon }}\beta(L_\theta) >0.$$ The virtual dimension is $$\mathrm{vdim}({\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)})=\beta(\det (T_W)) + (1-g)(\dim({W/\!\!/{{\bf G}}})-3)+k,$$ where $T_W\in K^\circ_{{\bf G}}(W)$ is the (virtual) ${{\bf G}}$-equivariant tangent bundle.

For each fixed class $\beta$ the set $\QQ_{>0}$ is divided into stability chambers by finitely many walls $1,\frac{1}{2},\dots,\frac{1}{\beta(L_\theta)}$ such that the moduli spaces remain constant in 