SOME EFFECTS OF PROPERTY Np ON THE HIGHER NORMALITY AND DEFINING EQUATIONS OF NONLINEARLY NORMAL VARIETIES

For a smooth projective variety X ⊂ P embedded by the complete linear system, Property Np has been studied for a long time([5], [11], [12], [10] 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 Np on higher normality and defining equations of a smooth variety embedded by sub-linear system. Also we prove a claim about Property Np of surface scrolls which is a generalization of Green’s work in [11] about Property Np for curves.