On Fujita's conjecture
Let \(L\) be an ample line bundle on a smooth \(n\)-fold \(X\) over an algebraically closed field of characteristic zero. It is conjectured that the adjoint bundle \(K_X+L\) is generated by global sections if \(L\) is ''very positive'' in some sense. The aim of this paper is to establish a higher dimensional version of Reider's famous result on surfaces. Main result: Let \(x\in X\) and assume \(L^n>n^n\) and \(L^d\cdot Z\geq m_x(Z)n^d\) for any subvariety \(Z\) with \(m_x(Z)\leq\binom{n-1}{d-1}\), where \(d=\dim Z<n\) and \(m_x(Z)\) is the multiplicity of \(Z\) at \(x\). Then \(K_X+L\) is globally generated at \(x\). The proof goes along the line pioneered by Ein-Lazarsfeld, and uses \textit{Y. Kawamata}'s idea [cf. ``On Fujita's freeness conjecture for 3-folds and 4-folds'', Math. Ann. 308, No. 3, 491-505 (1997)]. In case \(n=3\), by making use of an idea of the reviewer, the author shows further that \(K_X+L\) is free at \(x\) if \(L^3>27\), \(L^2S\geq 9\) for any surface \(S\ni x\) which is smooth or RDP at \(x\), and \(LC\geq 3\) for any curve \(C\) which is smooth at \(x\). He gives also a criterion for \(K_X+L\) to separate finite points.
scientific article; zbMATH DE number 1042480
scientific article