  Perturbative heterotic string theory develops a single complex tachyonic mode
beyond the Hagedorn temperature. We calculate the quartic effective potential
for this tachyonic mode at the critical temperature. Equivalently, we determine
the quartic effective potential for strong supersymmetric breaking via
anti-perdiodic boundary conditions for fermions on a small circle. We give many
details of the heterotic tachyon scattering amplitudes, including a unitarity
check to fix all normalization constants. We discuss difficulties in obtaining
an effective action valid at all radii. We argue that in certain variables, the
quartic term in the potential is radius independent. Speculations on the
properties of a new strongly curved phase that could occur after tachyon
condensation are offered.
