The Geometry of a Determinantal Variety I'm interested in the geometry of the set of all matrices whose rank is at most $k$. I've been told that this is referred to as a determinantal variety in algebraic geometry. This question is a bit awkward since it's being asked by someone who doesn't know algebraic geometry.
What is the geometry of a determinantal variety of rank at most $k$? As an example of what I mean consider a ball $B(0, 1) \subset \mathbb{R}^3$. I know that $B$ is convex, that it consists of one path-connected component, that it's boundary is a sphere, that the sphere has Gaussian curvature everywhere equal to 1, etc.
Is a determinantal variety of rank at most $k$ convex? How does convexity change as $k$ changes? How many path-connected components does it have? Where is the set of singularities? What's its curvature like?
 A: Let's work over $\mathbf{C}$. For $\mathfrak{gl}_2$ the only interesting example is rank $\le 1$:
$$ \left\{ \begin{pmatrix} h_1&e\\
f&-h_2\end{pmatrix}\ :\ ef+h_1h_2=0 \right\}$$
which is a cone, e.g. has a singular point at $0$, is connected, not convex and has  curvature $0$. These facts also hold for rank $\le n-1$ in $\mathfrak{gl}_n$.
In general, note that $\mathfrak{gl}_n$ has a $\text{GL}_n$ equivariant stratification by Jordan decomposition
$$\mathfrak{gl}_n\ =\ \coprod_{\lambda,\alpha} J(\lambda,\alpha)$$
where $\lambda\vdash n$ is a partition of $n$ giving the sizes of the Jordan blocks and $\alpha$ are the eigenvalues of the blocks. You can write each determinental variety as a union of certain $J(\lambda,\alpha)$'s, I won't do it explicitly but will talk more about the geometry. 
There is a partial order on partitions with $\lambda\le \mu$ iff $\lambda$ is a refinement of $\mu$. Then 
$$\overline{J(\lambda,\alpha)}\ =\ \coprod_{\lambda\le \mu}J(\mu,\alpha)$$
i.e. in the limit, blocks can merge, and nothing else. Moreover, $\text{GL}_n$ acts transitively on $J(\mu,\alpha)$ by conjugation, giving
$$J(\alpha,\lambda)\ =\ \text{GL}_n/L_\lambda$$
where $L_\lambda\simeq\prod \text{GL}_{\lambda_i}$ is the stabiliser of a given block. 
