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Let $X$ be a proper curve over a field $k$ with a node $P$. Suppose that $X \setminus \{P\}$ is smooth over $k$. Then there is a natural homomorphism $$ T_X\mathsf{Def} = \mathop{\mathrm{Ext}}\nolimits_{\mathcal O_X}^1(\Omega_X, \mathcal O_X) \to H^0(X, \mathop{\mathscr{Ext}}\nolimits_{\mathcal O_X}^1(\Omega_X, \mathcal O_X)). $$ Show that $H^0(X, \mathop{\mathscr{Ext}}\nolimits_{\mathcal O_X}^1(\Omega_X, \mathcal O_X))$ can be identified with the first-order deformations of $P$.

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    $\begingroup$ I think you'll find this in any book on deformation of singularities — try this. $\endgroup$
    – abx
    Dec 2, 2020 at 16:21

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