Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, that is proper and flat. Let $u:C \to X/G$ be a family of maps to a quotient stack ($X$ is a complex non-singular variety and $G$ is a reductive group). Given that the equivariant homology class $(u_s)_*[C_s] \in H^2_G(X)$ is constant for $s \in S \backslash \{0\}$, does $(u_0)_*[C_0]$ also represent the same homology class? Is this obvious?
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$\begingroup$ At the very least you want to assume $u$ is proper, or else $u^{-1}(0)$ could be empty. I guess if "curve" means irreducible, then you're not worried about $C$ having components lying entirely inside $u^{-1}(0)$, but I would usually exclude that by saying "flat". I guess $u_s : C_s \to X/G$, so the class you're asking about is $(u_s)_*[C_s]$ (not upper-star)? $\endgroup$– Allen KnutsonCommented Oct 17, 2013 at 16:20
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$\begingroup$ Thanks @AllenKnutson. I modified the question to reflect the changes you suggest. $\endgroup$– AnonCommented Oct 18, 2013 at 6:02
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