Ag. Algebraic Geometry – Families on Artin Rings and Deformations

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. **

Reference Request – Conjugation in orthographic Artin groups

I'm looking for a reference with the following result:

To let $ a $ and $ b $ two elements of a right-angled Artin group $ A $, Accept that $ a $ and $ b $ have a minimum length (in terms of the canonical generator set of $ A $) in their conjugation classes. To let $ a_1 cdots a_n $ and $ b_1 cdots b_m $ Represent words of minimal length $ a $ and $ b $ respectively. If $ a $ and $ b $ are conjugated in $ A $, then $ a_1 cdots a_n $ can be obtained from $ b_1 cdots b_m $ By applying the following operations: permutation of two consecutive letters that commute, and cyclic permutation.

I'm sure it's written somewhere, but I can not find where.