  In the context of $q$-Painlev\'e VI with generic parameter values, the
Riemann-Hilbert correspondence induces a one-to-one mapping between solutions
of the nonlinear equation and points on an affine Segre surface. Upon fixing a
generic point on the surface, we give formulae for the function values of the
corresponding solution near the critical points, in the form of complete,
convergent, asymptotic expansions. These lead in particular to the solution of
the nonlinear connection problem for the general solution of $q$-Painlev\'e VI.
We further show that, when the point on the Segre surface is moved to one of
the sixteen lines on the surface, one of the asymptotic expansions near the
critical points truncates, under suitable parameter assumptions. At
intersection points of lines, this then yields doubly truncated asymptotics at
one of the critical points or simultaneous truncation at both.
