# Smoothness of moduli spaces of stable maps

If $$X$$ is a projective variety the moduli space of stable maps $$\overline{M}_{0,0}(X,\beta)$$ is a normal variety with finite quotient singularities.

Can the pairs $$(X,\beta)$$ such that $$\overline{M}_{0,0}(X,\beta)$$ is smooth as a variety be characterized?

I know that $$\overline{M}_{0,0}(\mathbb{P}^2,2)$$ is smooth. For instance, are $$\overline{M}_{0,0}(G(2,4),2)$$ and $$\overline{M}_{0,0}(LG(2,4),2)$$ smooth? Here $$G(2,4)$$ is the Grassmannian of planes in $$\mathbb{C}^4$$ and $$LG(2,4)\subset G(2,4)$$ the Lagrangian Grassmannian.

Thank you very much.

Here is a partial answer to your question. If $$X$$ is homogeneous then $$\overline{M}_{0,0}(X,\beta)$$ is a projective normal variety with at most finite quotient singularities. The singularities arise along the loci parametrizing maps with non trivial automorphisms. However, if such a locus is in codimension one the general point of the divisor is a smooth point of $$\overline{M}_{0,0}(X,\beta)$$ since it is locally at such a point the quotient of a smooth variety by a group generated by pseudoreflections.
For instance, consider in $$\overline{M}_{0,0}(G(2,4),2)$$ the locus $$\Gamma$$ of maps that are $$2$$ to $$1$$ onto a line. The Fano variety of lines in $$G(2,4)$$ is $$5$$-dimensional and for any line you have to choose two points on it identifying the ramification of the map. So $$\Gamma$$ has codimension $$2$$ in $$\overline{M}_{0,0}(G(2,4),2)$$ and since the general map in $$\Gamma$$ has a non-trivial involution $$\overline{M}_{0,0}(G(2,4),2)$$ is singular along $$\Gamma$$.
A similar argument shows that the locus in $$\overline{M}_{0,0}(LG(2,4),2)$$ parametrizing $$2$$ to $$1$$ maps onto a line is a divisor. So $$\overline{M}_{0,0}(LG(2,4),2)$$ is smooth.