Some effects of property N_p on the higher normality and defining equations of nonlinearly normal varieties

For a smooth projective variety X ⊂ ℙ^r embedded by the complete linear system, Property N_p has been studied for a long time ([5], [11], [12], [7] etc.). On the other hand, Castelnuovo-Mumford regularity conjecture and related problems have been focused for a projective variety which is not necessarily linearly normal ([2], [13], [15], [17], [20] etc.). This paper aims to explain the influence of Property N_p on higher normality and defining equations of a smooth variety embedded by a sub-linear system. Also we prove a claim about Property N_p of surface scrolls which is a generalization of Green's work in [11] about Property N_p for curves.