Automorphisms Deformations Obstructions Of Stable Curves

Proposition

[Prop. 5.3.23 in [Alp24]] Let \(A'\to A\) be surjection of noetherian local rings with residue field \(k\). Suppose that \(J = \ker(A'\to A)\) satisfies \(\mathfrak m_{A'}J = 0\). Let \((\mathcal C\to \Spec A, \sigma_{i})\) be a family of prestable curves over \(A\), and let \((C,p_i)\) be its base change to the residue field \(k\). Let \(D = \sum_i p_i\) be the marked point divisor of \(C\).

1. The group of automorphisms of a deformation of \((\mathcal C\to \Spec A, \sigma_i)\) over \(A'\) is bijective to $ \Ext⁰_{\mathcal O_C}(Γ_C(D), \mathcal O_C⊗_kJ)$
2. If there exists a deformation of \((\mathcal C\to \Spec A, \sigma_i)\) over \(A'\), then the set of isomorphism classes of all such deformations is a torsor under \(\Ext^{1}_{\mathcal O_C}(\Omega_{C}(D), \mathcal O_C\otimes_{k}J)\).
3. There is an element \(\operatorname{ob}_{\mathcal C} \in \Ext^2_{\mathcal O_C}(\Omega_C(D), \mathcal O_C \otimes_{k} J)\) with the property that there exists a deformation of \((\mathcal C\to \Spec A, \sigma_i)\) if and only if \(\operatorname{ob}_{\mathcal C} = 0\).