Is there any singular compactification of $PGL(2,\mathbb{C})$? Only compactification is known to me is $\mathbb{P}^3$. Does anyone know any other compactification?



Compactification of $PGL(2,\mathbb{C})$
The answer depends on how much symmetry of $PGL(2)$ you want to retain.


If the compactification is to be equivariant for left and right multiplication then, indeed, $\mathbb P^3$ is the only normal compactification. This follows easily from the embedding theory of spherical varieties.


If no symmetry is to be preserved then simply blow up up $\mathbb P^3$ in some crazy curve in the boundary and you obtain with probabilty one a singular variety.


Interesting is the case where only left but not right multiplication extends to the compactification. Here, the group $SL(2)$ has been studied extensively but the results for $PGL(2)$ are pretty much the same (just devide by the center).


On of the first systematic investigation in theis direction is "Quasihomogeneous affine algebraic varieties of the group SL(2)" by V.L. Popov. He classifies normal *affine* $SL(2)$-varieties on which $SL(2)$ has an open orbit. If the isotropy group of the generic orbit is finite then pretty much none of these spaces is smooth.


A typical example is the orbit closure of a nullform $x^df(x,y)$ where $f$ is homogeneous of degree $<d$. Note that all of these spaces can be equivariantly compactified.


Later Luna-Vust classified *all* left equivariant embeddings of $SL(2)$. There are plenty. Lucy Moser determined all normal embeddings of $SL(2)/\Gamma$ with $\Gamma$ finite. In particular $PGL(2)$ is covered.


