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Given a nodal (= reduced, connected, projective, having only ordinary double points as singularities) curve $C$ consisting of 5 $\mathbb P^1$ labeled $C_0, D_1, D_2, D_3, D_4$ such that $D_i$ intersects only $C_0$ in exactly one point $P_i$ and $P_i \neq P_j$ for $i \neq j$ transversally. The dual graph of $C$ is a cross with $C_0$ in the middle and $D_1, \ldots, D_4$ as leafs.

I want to compute the dimension dim $H^0(C, T_C)$ of the global sections of the tangent sheaf $T_C$ to $C$?

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What do you mean by tangent bundle of a singular variety ? – user18119 Jun 9 '12 at 20:15
@QiL: Good point. Instead of "tangent bundle" I should have used "tangent sheaf", that is $T_C = \mathcal Hom_{\mathcal O_C}(\Omega_C, \mathcal O_C)$, where $\Omega_C$ is the sheaf of differentials on $C$ . In fact, $T_C$ is locally free iff $C$ is non-singular. – boxdot Jun 10 '12 at 9:21
up vote 4 down vote accepted

I've found the solution in Hartshorne's book Deformation Theory on p. 183. I formulate the solution for an arbitrary nodal curve $C$ consisting of irreducible compontents $C_i \cong \mathbb P^1$.

Let $S$ be the set of singular points in $C$. Locally $C$ around each node in $S$ the curve looks like $(xy=0) \subset \mathbb A^2$. Hence, locally, $T_C = xT_D \oplus yT_{D'}$, where $D, D'$ are the components through the chosen node. It follows, globally, we have $$ T_C \cong \bigoplus_{C_i \subset C} (\mathcal I_{S\cap C_i} \otimes T_{C_i}), $$ where $\mathcal I_{S\cap C_i}$ is the sheaf of ideals in $\mathcal O_{C_i}$ which defines the closed subscheme $S \cap C_i$ of $C_i$ consisting of nodes in $C_i$. Now $h^0(C_i, \mathcal I_{S\cup C_i} \otimes T_{C_i}) = max(3 - \#(S\cap C_i), 0)$ which follows from $C_i \cong \mathbb P^1$.

So the answer in the case of the above curve is $h^0 = 8$.

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