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Fialowski and Schlichenmaier constructed examples of global deformations ofLie algebras of vector fields from deforming the underlying variety. Weformulate their approach in a conceptual way. Namely, we construct a stack ofdeformations and a morphism form the moduli stack of stable marked curves. Themorphism associates to a family of marked curves the family of Lie algebrasobtained by taking the Lie algebra of vertical vector fields on the familywhere one has extracted the marked points. We show that this morphism is almosta monomorphism by Pursell-Shanks theory.

Deformations of Lie algebras of vector fields arising from families of schemes