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

The goal of the present paper is to construct examples of global deformations of vector field Lie algebras in a conceptual way. Fialowski and Schlichermaier [8] constructed global deformations of the infinitesimally and formally rigid Lie algebra of polynomial vector fields on the circle and of its central extension, the Virasoro algebra. The Lie algebra of polynomial vector fields on the circle is here replaced by/seen as the Lie algebra of meromorphic or rational vector fields on the Riemann sphere admitting poles only in the points 0 and ∞. In this context one gets non trivial deformations from an affine family of curves by first deforming it as a projective family with marked points and then extracting the points. Fialowski and Schlichermaier get in this way non-trivial deformations of Lie algebras, and the underlying families of curves present singularities. In the attempt of producing deformations of vector field Lie algebras from deformations of the underlying pointed algebraic variety in a general framework, we are led to a notion of global deformation which is different from the one used by Fialowski and Schlichenmaier, one which is closer to deformations in algebraic geometry. A first goal is to compare these global deformations with the corresponding notion from Fialowski-Schlichermaier’s article. We then stick to the notion of deformations of the Lie algebra of vector fields on a pointed algebraic curve imposed by the deformation of the underlying curve, and show its close relation to the moduli space of pointed curves. In order to formulate this relation, we show that the “space of deformations” carries the geometric structure of a C-stack. This is a kind of functor from the category of model spaces, here affine schemes, to the category of groupoids (in order to capture the idea of considering deformations up to isomorphism). It has to satisfy conditions to provide construction of geometric objects by gluing local data, making it ressemble the functor of points of a scheme. The link between the moduli stack Mg,n and the deformation stack Def is a morphism of stacks I. It realizes a family of marked projective curves as a deformation of the Lie algebra of regular vector fields on the affine curve obtained from extracting the