Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow \mathcal{M}_{g}$. As such, one has the natural short exact sequence
$0 \rightarrow T_{C} \rightarrow T\mathcal{M}_{g,1}|_C \rightarrow N_C \rightarrow 0$
where $T_C$ and $N_C$ are tangent and normal bundles of $C$. As usual, there is an associated long exact sequence of cohomology:
$0 \rightarrow H^0(C, T_{C}) \rightarrow H^0(C,T\mathcal{M}_{g,1}|_C) \rightarrow H^0(C, N_C) \rightarrow H^1(C, T_{C}) \rightarrow H^1(C,T\mathcal{M}_{g,1}|_C) \rightarrow H^1(C, N_C) \rightarrow 0$
Some easy facts:
- $H^0(C, T_{C}) = 0$ for $g > 1$.
- $H^1(C, TC) \simeq \mathbb{C}^{3g - 3}$.
- $N_C \simeq H^1(C, TC) \otimes \mathcal{O}_C$.
- $H^0(C, N_C) \simeq H^1(C, TC) \otimes \mathbb{C}$.
- $H^1(C, N_C) \simeq H^1(C, TC) \otimes \mathbb{C}^{g}$.
Two (related) questions:
1) Is the normal bundle sequence split?
2) What is the dimension of $H^0(C, T\mathcal{M}_{g,1}|_C)$?