On Higher Syzygies of Projective Toric Varieties

Let A be an ample line bundle on a projective toric variety X of dimension n (≥ 2). It is known that the d-th tensor power A⊗d embedds X as a projectively normal variety in Pr := P(H0(X,L⊗d)) if d ≥ n − 1. In this paper first we show that when dimX = 2 the line bundle A⊗d satisfies the property Np for p ≤ 3d − 3. Second we show that when dimX = n ≥ 3 the bundle A⊗d satisfies the property Np for p ≤ d − n + 2 and d ≥ n − 1.