  A canonically-embedded curve of genus $g$ is a pure 1-dimensional,
non-degenerate subscheme $C$ of ${\bf P}^{g-1}$ over an algebraically closed
field $k$, for which ${\cal O}_C(1) \cong \omega_C$, (the dualizing sheaf)$ and
$h^0(C, {\cal O}_C) = 1$, $h^0(C, \omega_C) = g$. The singularities of $C$ (if
any) are Gorenstein, and $C$ is connected of degree $2g-2$ and arithmetic genus
$g$. In a recent paper, Schreyer has proved that Petri's normalization of the
homogeneous ideal $I(C)$ of a smooth canonically-embedded curve can be also
carried out for singular curves, provided that the curve has a simple
$(g-2)$-secant (a linear ${\bf P}^{g-3}$ intersecting $C$ transversely at
exactly $g-2$ (smooth) points). We use the Petri normalization to study the
Hilbert scheme of curves of degree $2g-2$ and arithmetic genus $g$ in ${\bf
P}^{g-1}$ in the low-genus cases $g = 5,6$. The main results are that the
Hilbert points of all curves for which Petri's approach applies lie on one
irreducible component of the Hilbert scheme in these cases. (The same is not
true for $g$ sufficiently large.)
