Global generation of adjoint bundles

Abstract

In 1988, I. Reider proved that for a smooth projective surface Xand an ample line bundle Lon X, K _x+ 3L is globally generated and K _x + 4L is very ample ([12]). In fact his theorem is much stronger than this (see [12] for detail). Recently a lot of results have been obtained about effective base point freeness (cf. [1, 3, 8, 13, 14, 15]). In particular J. P. Demailly proved that 2K _X + 12n ⁿL is very ample for a smooth projective n-fold Xand an ample line bundle Lon X. [2] will give a good overview for these recent results. The motivation of these works is the following conjecture posed by T. Fujita.