Arithmetically Cohen–Macaulay algebraic curves ∗

Let C be a numerically connected curve lying on a smooth algebraic surface. We show that an invertible sheaf H num ∼ ωC ⊗ A is normally generated on C if A is an ample invertible sheaf of degree ≥ 3. As a corollary we show that on a smooth algebraic surface of general type the invertible sheaf K⊗3S yields a projectively normal embedding of S assuming KS ample, (KS) 2 ≥ 3, pg(S) ≥ 2 and q(S) = 0