Let $k$ be a field $\mathbb C$.
Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$.
The collection $X$ of matrices with repeated eigenvalues over $\overline k$ is a subvariety (as it is the zero set of the discriminant of the characteristic polynomial), and moreover it is preserved by $G$.
If we let $k^n\subset X$ be the diagonal matrices with repeated roots, then $Y=X\setminus G(D)$ is the set of non-diagonalizable matrices, and also has an action of $G$.
If $k=\overline k$, then every $G$-orbit contains an element in Jordan normal form, and by scaling the off-diagonal entries, we remain in the same conjugacy class, and so we see the corresponding diagonal matrix is in the closure of the orbit.
Therefore $Y$ is dense in $X$. This allows one to compute the dimension of $Y$ (I think).
However, I'm not really sure what else to say in describing $Y$.
What does $Y$ look like? I know this is a little vague, but I'm not really sure what a reasonable reformulation would be.
Are there good decompositions of $Y$ that help in understanding its structure? Is it smooth? Is it a manifold? Can we calculate useful invariants of $Y$, such as the cohomology? Are we better off understanding the individual orbits?
Are there other group actions on $Y$ which elucidate its structure?