## 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.