Groebner techniques for low degree Hilbert stability

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.