Let C be a numerically connected curve lying on a smooth algebraic surface. We show that an invertible sheaf H num. eq. to K_C +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 3K_S yields a projectively normal embedding of S assuming K_S ample, K_S^2 >=3, pg(S) >= 2 and q(S) = 0.
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