Deformation theory

Summary

Deformation theory studies infinitesimal variations of a geometric or algebraic object: families over small base spaces whose special member is the given object. In algebraic geometry the small base spaces are spectra of local Artinian rings, and the first step is the ring of dual numbers $k[\varepsilon]/(\varepsilon^2)$. For a smooth variety, first-order deformations are classified by the first cohomology group of the tangent sheaf. The obstructions to extending them lie in the second cohomology group. It describes the local structure of moduli spaces near a point.