For example, if H $H$ is a hyperplane, then P^n $\mathbb{P}^n - H = A^n, \mathbb{A}^n$, which is a vector space.
If $n = m^2 - 11$, then we can regard A^n+1 $\mathbb{A}^{n+1}$ as the space of m x $m \times m$ matrices and take the hypersurface H $H$ in P^n $\mathbb{P}^n$ corresponding to the singular matrices. The complement P^n $\mathbb{P}^n - H H$ is PGL_n.$\mathbf{PGL}_n$.
If we restrict ourselves to irreducible H, $H$, are there any more examples besides the two above?
If we allow reducible hypersurfaces, then we can get a few more. We can realize the multiplicative group Gm $\mathbb{G}_m$ as P^1 $\mathbb{P}^1$ minus two points, and removing the union of two distinct lines from P^2 $\mathbb{P}^2$ will give us Gm x A^1. $\mathbb{G}_m \times \mathbb{A}^1$. What can we say about the situation here?
The complement of a hypersurface is affine, so only linear algebraic groups will arise.
I haven't put much thought into the base field, so we can just start with C.$\mathbb{C}$.
