On syzygies of Veronese embedding of arbitrary projective varieties

For a very ample line bundle L on a projective scheme X, let φLℓ:X↪PH0(X,Lℓ), ℓ⩾1, be the embedding defined by the complete linear series |Lℓ|. In this paper we study the problem how the Castelnuovo–Mumford regularity of φL(X) effects on the defining equations of φLℓ(X) and the syzygies among them. We show that if φL(X)⊂PH0(X,L) is m-regular, then (X,Lℓ) satisfies property N2ℓ−m+1 for , and (X,Lℓ) satisfies property Nℓ for all ℓ⩾m−1.