Let $X$ and $Y$ be two subvarieties of $n\times n$ matrices. My question is that is there any condition to guarantee that there exits some matrix $g$ such that $Y=g^{-1} X g$? If such $g$ exists, then let's say $X$ is conjugate to $Y$. It is clear that the necessary condition is that $\dim X=\dim Y$ and $\deg X= \deg Y$.

The motivation of this problem is that if $X,Y$ are just two points, then we can use Jordan canonical form to check whether or not $X$ is conjugate to $Y$. But what if $X,Y$ are of higher dimension? For example, if $X,Y$ are two curves in the space of $n\times n$ matrices, when is $X$ conjugate to $Y$? Another simple case is that if $X,Y$ are linear subspaces of dimension 2, then when is $X$ conjugate to $Y$? In this case, one can rephrase the question as follows, consider the action of $GL_n$ on the Grassmannian $Gr(2,n^2)$ by conjugation. When are two points of $Gr(2,n^2)$ in the same orbit under this action?