I’m reading the paper 13/2 Ways of Counting Curves by Pandharipande and Thomas. I’m very confused with the following statement on page 8
$\S$ Deformation theory. We return now to the deformation theory for embedded curves briefly discussed in Section $\frac{1}{2}$. The deformation theory for arbitrary stable maps is very similar.
Let $C \subset X$ be a nonsingular embedded curve with normal bundle $\nu_C$. The Zariski tangent space to the moduli space $\overline{\mathcal{M}}_g(X,\beta)$ at the point $[C \to X]$ is given by $H^0(C, \nu_C)$. Locally, we can lift a section of $\nu_C$ to a section of $T_X |_C$ and deform $C$ along the lift to first order. Since globally $\nu_C$ is not usually a summand of $T_X|_C$ but only a quotient, the lifts will differ over overlaps by vector fields along $C$. The deformed curve will have a complex structure whose transition functions differ by these vector fields. In other words, from
$$0 \to T_C \to T_X |_C \to \nu_C \to 0$$
we obtain the sequence
$$(1.1) \quad 0 \to H^0(C, T_C) \to H^0(C, T_X |_C) \to H^0(C, \nu_C) \to H^1(C, T_C)$$
How do I understand the Zariski tangent space? Many thanks!
Best Answer
Let's say that our moduli space $\mathcal{M}$ is a scheme. In that case, for a given point $x\in \mathcal{M}$, the Zariski tangent space $T_{x}\mathcal{M}$ is the usual tangent space, i.e. given by $(\mathfrak{m}_x/\mathfrak{m}_x^2)^\vee$, where $\mathfrak{m}_x$ is the unique maximal ideal of the local ring $\mathcal{O}_{\mathcal{M},x}$. The tangent directions to $x$ represent first order deformations of $x$ in the ambient space, $\mathcal{M}$.
The authors are making the claim that there is a canonical identification of vector spaces between the tangent space to $\overline{\mathcal{M}}_g(X;\beta)$ at the point $[C\to X]$ and $H^0(C,\nu_C)$. The intuition for this is as follows: if we consider deforming a single point $x\in X$ to first order in $X$, we consider an element of $T_x X$. In the case that we are deforming a family of points (e.g. a curve $C$ in $X$), we should choose a vector at each point $x\in C$. This gives us a vector field along $C$, i.e. an element of $H^0(C,T_X|_C)$. However, to ensure we are actually deforming into the ambient directions (as opposed to defining a vector field which flows the curve into itself), we should really consider vector fields that point outwards from $C$. That is, we should consider elements of $H^0(C,\nu_C)$.
This is not a proof, but at least gives some intuition for this idea.
Also, a technical comment. In this case, $\overline{\mathcal{M}}_g(X;\beta)$ should really be a (Deligne-Mumford) stack. There is a functorial definition of the Zariski tangent space in terms of maps $$\mathrm{Spec}\:k[\epsilon]/(\epsilon^2) \to \overline{\mathcal{M}}_g(X;\beta)$$ better suited to this case. For this, you can see Definition 3.5.7 of Alper (n.b. the numbering may change as the text is updated often).