On syzygies of Veronese embedding of arbitrary projective varieties

For a very ample line bundle L on a projective scheme X, let φ_Lℓ : X {right arrow, hooked} P H⁰ (X, L^ℓ), ℓ ≥ 1, be the embedding defined by the complete linear series | L^ℓ |. In this paper we study the problem how the Castelnuovo-Mumford regularity of φ_L (X) effects on the defining equations of φ_Lℓ (X) and the syzygies among them. We show that if φ_L (X) ⊂ P H⁰ (X, L) is m-regular, then (X, L^ℓ) satisfies property N_{2 ℓ - m + 1} for frac(m, 2) ≤ ℓ ≤ m - 2, and (X, L^ℓ) satisfies property N_ℓ for all ℓ ≥ m - 1.