Deformations of pointed stable maps with "curve held rigid" or "preserving the dual graph"

Question

I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order deformations of a pointed stable map $\mu = (C, p_i, \mu: C \to X)$. For $C = \mathbb P^1$ they write down an exact sequence $$ 0 \to H^0(C,T_C) \to Def_R(\mu) \to Def(\mu) \to 0, $$ where $Def_R(\mu)$ is the space of first order deformations of $\mu$ with $C$ held rigid.

What does they mean by held rigid?

Later $C$ is a tree of $\mathbb P^1$'s. Let $G$ be the dual graph of $C$. Now they consider the space $Def_G(\mu) \subset Def(\mu)$ of first order deformations of the pointed stable map $\mu$ preserving the dual graph.

What does they mean by preserving the dual graph?

If I understand it correctly, $Def(\mu)$ consists of all $(\mathcal C \to Spec\ \mathbb C[\varepsilon], \bar p_i, \bar \mu: \mathcal C \to X)$ such that $\mathcal C \otimes_{\mathbb C[\varepsilon]} \mathbb C \cong C, \bar p_i \otimes \mathbb C = p_i$ and $C \to \mathcal C \xrightarrow{\bar \mu} X$ coincides with $\mu$. So, a geometric fiber of $\mathcal C$ should have the same dual graph as $C$.

Thanks a lot.

Answer 1

Dear finite,

First, a confusing typo in your short exact sequence. It should be $$ 0 \to H^0(C,T_C) \to \mathrm{Def}_R(\mu) \to \mathrm{Def}(\mu) \to 0.$$ What they mean by "held rigid" here is that they consider maps from $C = \mathbf P^1$ with a fixed choice of coordinates, not up to automorphism of $\mathbf P^1$. If we ignore marked points then the more "global" way of saying this is that $\mathrm{Def}(\mu)$ is the tangent space of $\overline{M}_{0,0}(X,\beta)$ at the point $[\mu]$, whereas $\mathrm{Def}_R(\mu)$ is the tangent space of the Hom scheme $\mathrm{Hom}(C,X)$, which just parametrizes morphisms $C \to X$, at $[\mu]$. Naively, one expects then that $\overline{M}_{0,0}(X,\beta)$ is locally the quotient of $\mathrm{Hom}(C,X)$ by $\mathrm{Aut}(C) \cong \mathrm{PGL}(2)$. This is essentially what the short exact sequence expresses, note e.g. that $\dim H^0(C,T_C) = \dim \mathrm{PGL}(2) = 3$.

In the definition of $\mathrm{Def}_G(\mu)$ they just mean that they consider first order deformations whose dual graph is $G$. The corresponding "global" way of saying this is that they consider the locally closed substack of $\overline{M}_{0,n}(X,\beta)$ defined by demanding that the dual graph is $G$, and consider the tangent space of this substack.