Let's work with a class of schemes about an algebraically closed field $ k $ so that any two schemas in this class are isomorphic.

An example of such a class would not be singular curves of the genus zero $ k $, To let $ X $ occur as a representative of this class.

Is the set of isomorphism classes of such families over an artin ring $ A $ equivalent to the set of isomorphism classes deformations of $ X $ over $ A $?

The only problem with checking such a statement is that we allow families $ mathcal {X} $ over $ B $ have different fibers, while a deformation fixes the fiber.

** Please accept every beautiful feature you desire. I did not want to specify the class as non-singular zero curves because I do not want to $ H ^ 1 (X, mathcal {T} X) = (0) $I am interested in a scheme with non-trivial deformations. **