The topology of the relative character varieties of a quadruply-punctured sphere

Let M be a quadruply-punctured sphere with boundary components A, B, C, D. The SL(2, C)-character variety of M consists of equivalence classes of homomorphisms rho of pi(1)(M) --> SL(2, C) and can be identified with a quartic hypersurface in C-7. For fixed a, b, c, d is an element of C, the subset V-a,V-b,V-c,V-d corresponding to representations rho with tr(p(A))= a, tr(rho(B)) = b, tr(rho(C)) = c, tr(rho(D)) = d is a cubic surface in C-3. We determine the singular points of Va,b,c,d and classify its set V-a,V-b,V-c,V-d(IR) of IR-points into six topological types, at least when this set is nonsingular. V-a,V-b,V-c,V-d(IR) contains a compact component if and only if -2 < a, b, c, d < 2. For certain values of (a, b, c, d), this component corresponds to representations in SL(2, IR).