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Let $X$ be a $\mathbb{C}^*$-equivariant algebraic variety. Then there is a notion of a map to $X$ twisted by a line bundle. Namely, let $B$ be a variety and $L/B$ a line bundle. Let $P_L=L\setminus B$ be the principal $\mathbb{C}^*$ bundle defined by $L$. Given this data, define $\operatorname{Maps}_L(B,X)$ to be the space of $\mathbb{C}^*$-equivariant maps $P\to B$. When $L$ is the tirival line bundle, this is of course the same thing as the space of maps $B\to X$.

Now let's assume that $B$ is a nodal curve with a line bundle $L$. I want to know whether there is a notion of stability for $f\in \operatorname{Maps}_L(B,X)$ which reduces to the usual stability condition on $\overline{\mathcal{M}}_{g,n}(X)$ for trivial $L$? And when $B$ and $L$ vary, do these spaces form a decent moduli family?

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    $\begingroup$ This sounds quite similar to the work of Martens-Thaddeus, albeit in genus $0$. This also sounds similar to the recent work of Bumsig Kim. $\endgroup$ – Jason Starr Apr 28 '15 at 8:22

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