  For a smooth projective variety $X\subset \P^r$ embedded by the complete
linear system, Property $N_p$ has been studied for a long time. On the other
hand, Castelnuovo-Mumford regularity conjecture and related problems have been
focused for a projective variety which is not necessarily linearly normal. This
paper aims to explain the influence of Property $N_p$ on higher normality and
defining equations of a smooth variety embedded by sub-linear system. Also we
prove a claim about Property $N_p$ of surface scrolls which is a generalization
of Green's work in \cite{Green} about Property $N_p$ for curves.
