On the projective normality of the adjunction bundles. --- Appendix (by M. Andreatta, E. Ballico and A. J. Sommese) (Q1179510)

Let \(X\) be a smooth projective variety of dimension \(n\), polarized by a very ample line bundle \(L\). Let \(K_ X\) be the canonical bundle and let \({\mathcal L}_{(a,b)}:=K^ a_ X\otimes L^ b\). Assume that \((X,L)\) is neither \((\mathbb{P}^ n,{\mathcal O}(1))\), \((\mathbb{P}^ 2,{\mathcal O}(2))\), \((Q,{\mathcal O}(1))\), \(Q\) hyperquadric in \(\mathbb{P}^{n+1}\), or a \(\mathbb{P}^{n-1}\) bundle over a smooth curve with the restriction of \(L\) to a fibre isomorphic to \({\mathcal O}(1)\). Then from results of Sommese and Van de Ven it follows that the map associated to \(\Gamma({\mathcal L}_{(a,b)})\) is an embedding. In the paper the following is proved: Theorem. Suppose that \((X,L)\) is not one of the pairs of the previous list. If \(a\geq 1\) and \(b>(n-1)a+1\), then \((X,{\mathcal L}_{(a,b)})\) is projectively normal. A pair \((X,L)\) is said projectively normal if the natural maps \(S^ \rho H^ 0(X,L)\to H^ 0(X,L^ \rho)\) are surjective for \(\rho\geq 1\). In the appendix by the same authors and \textit{E. Ballico} the previous result is extended to the case \(b=(n-1)a+1\). Note that for \(a=1\), the bound on \(b\) is sharp. The question on the projective normality of \(K_ X\otimes L^ b\) with \(b\geq 2\) and \(n=2\) was posed by S. Mukai and M. L. Green. --- L. Ein and R. Lazarsfeld independently obtained some results related to the result above.