Projectively normal adjunction surfaces

Abstract

On a projective surface S S polarized by a very ample line bundle L L one can consider the adjoint bundles ( K S ⊗ L ) ⊗ n = L ⊗ n {({K_S} \otimes L)^{ \otimes n}} = {L^{ \otimes n}} and the adjunction mappings associated to them. Suppose these mappings are embeddings (it is well known when this is the case: see [So-VdV]). We prove that these embeddings are projectively normal for n ≥ 2 n \geq 2 and we describe some counterexamples for n = 1 n = 1 . For n ≥ 2 n \geq 2 we show that the ideals of the image of S S are generated by quadrics.