  This paper is a sequel to arXiv:1109.4986, where we proved that a general
smooth curve of odd genus, canonically or bicanonically embedded, has
semistable finite Hilbert points. Here, we prove that a generic canonically
embedded curve of even genus has semistable finite Hilbert points. More
precisely, we prove that a generic canonically embedded trigonal curve of even
genus has semistable finite Hilbert points. Furthermore, we show that the
analogous result fails for bielliptic curves. Namely, the Hilbert points of
bielliptic curves are asymptotically semistable but become non-semistable below
a definite threshold value depending on genus.
