  We give a method for verifying, by a symbolic calculation, the stability or
semistability with respect to a linearization of fixed, possibly small, degree
$m$, of the Hilbert point of a scheme $X \in {\mathbb P}(V)$ having a suitably
large automorphism group. We also implement our method and apply it to analyze
the stability of bicanonical models of certain curves. Our examples are very
special, but they arise naturally in the log minimal model program for
$\bar{\mathcal M}_g$. In some examples, this connection provides a check of our
computations; in others, the computations confirm predictions about conjectural
stages of the program.
