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# For which hypersurfaces H in P^nprojectivespace does the complement P^n-H admit an algebraic group structure?

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.$\mathbb{C}$.

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# For which hypersurfaces H in P^n does the complement P^n - H admit an algebraic group structure?

For example, if H is a hyperplane, then P^n - H = A^n, which is a vector space.

If n = m^2 - 1, then we can regard A^n+1 as the space of m x m matrices and take the hypersurface H in P^n corresponding to the singular matrices. The complement P^n - H is PGL_n.

If we restrict ourselves to irreducible 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 as P^1 minus two points, and removing the union of two distinct lines from P^2 will give us Gm x 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.