Representation stability for the Kontsevich space of stable maps

For a fixed algebraic variety $X$, curve class $α\in N_1(X)$, and genus $g \in \mathbb N$, we consider the sequence of $S_n$ representations obtained from the homology of the Kontsevich space of stable maps to $X$, $\bar M_{g,n}(X,α)$. Using the category of finite sets and surjections, we prove a representation stability theorem that governs the behavior of this sequence of representations for $n$ sufficiently large.

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