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Vietnam Journal of Mathematics 35:2(2007) 135-151  Adjoint Line Bundles and Syzygies of Projective Varieties Huy Tai Ha Abstract.  Let $X$ be a smooth projective variety and let $K$ be the canonical divisor of $X$. In this paper, we study embeddings of $X$ given by adjoint line bundles $K \otimes L$, where $L$ is an ample invertible sheaf. When $X$ is a regular surface and $K \otimes L$ is normally generated (i.e. satisfies property $N_0$), we obtain a numerical criterion for $K \otimes L$ to have property $N_p$. When $X$ is a regular variety of arbitrary dimension, under a mild condition, we give an explicit calculation for the regularity of ideal sheaves of such embeddings.   2000 Mathematics Subject Classification: 14F17, 13D02, 14J60, 14C20. Keywords: Adjoint line bundles, syzygies, minimal free resolution, Koszul cohomology.

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