Let $\mathcal{X}\to S$ be a smooth family of projective varieties. Let $\beta$ be a curve class supported in some fiber and consider the relative moduli space of stable maps $\overline{M}_{g,n}(\mathcal{X},\,\beta)\to S$. Is the virtual fundamental cycle $[\overline{M}_{g,n}(\mathcal{X},\,\beta)]^{\rm vir}$ represented by a linear combination of algebraic cycles which are flat over $S$? Note that this would imply deformation-invariance of the GW invariant when these cycles have relative dimension $0$ over $S$.

If anyone has a citation, that would be very useful!

Let $\mathcal{X}\to S$ be a smooth family of projective varieties over a smooth curve $S$. Let $\beta$ be a curve class supported in some fiber and consider the relative moduli space of stable maps $\overline{M}_{g,n}(\mathcal{X},\,\beta)\to S$. Is the virtual fundamental cycle $[\overline{M}_{g,n}(\mathcal{X},\,\beta)]^{\rm vir}$ represented by a linear combination of algebraic cycles which are flat over $S$? Note that this would imply deformation-invariance of the GW invariant when these cycles have relative dimension $0$ over $S$. I am also happy to consider the case where the moduli space has dimension zero over the general fiber of $S$ and the invariant is enumerative. In this case, I'm simply asking whether the closure of the moduli space of the general fiber represents the virtual fundamental class of a special fiber.

If anyone has a citation, that would be very useful!