  We study the component structures of some standard-graded Hilbert schemes
closely related to a Hilbert scheme of curves studied by Gotzmann. In
particular, we encounter examples of singular lex-segment points lying on two
and three irreducible components. We find further singular lex-segment points
at nearby Hilbert schemes. We conclude by showing that the analogous example at
the Hilbert scheme of twisted cubics also has a singular lex-segment point.
