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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 $H^0(C, T\mathcal{M}_{g,1}|_C)$?

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    $\begingroup$ The sequence is maximally nonsplit: i.e., the connecting map from $H^1(C,T_C)$ to $H^1(C,T_C)$ is the Kodaira-Spencer map, and it is an isomorphism (at least if you work with $\mathcal{M}_g$ as an orbifold, e.g., if $C$ has no non-identity automorphisms). $\endgroup$ May 5, 2016 at 21:09

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