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

Fialowski and Schlichenmaier constructed global deformations of the infinitesimally and formally rigid Lie algebra of polynomial vector fields on the circle from families of projective curves with marked points. In the present article, we show how to obtain these examples in a conceptual way. For this, we define a stack of deformations of Lie algebras and study the morphism from the moduli stack to this deformation stack which associates to a family of marked curves the Lie algebra of vertical vector fields on the punctured family.