  It is shown that the representation theory of some finitely presented groups
thanks to their $SL_2(\mathbb{C})$ character variety is related to algebraic
surfaces. We make use of the Enriques-Kodaira classification of algebraic
surfaces and the related topological tools to make such surfaces explicit. We
study the connection of $SL_2(\mathbb{C})$ character varieties to topological
quantum computing (TQC) as an alternative to the concept of anyons. The Hopf
link $H$, whose character variety is a Del Pezzo surface $f_H$ (the trace of
the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic
state computing, derived from the trefoil knot in our previous work, may be
seen as TQC from the Hopf link. The character variety of some two-generator
Bianchi groups as well as that of the fundamental group for the singular fibers
$\tilde{E}_6$ and $\tilde{D}_4$ contain $f_H$. A surface birationally
equivalent to a $K_3$ surface is another compound of their character varieties.
