A tight closure proof of Fujita's freeness conjecture for very ample line bundles
The main result of this paper is theorem 2, which says that for a globally generated ample line bundle \(L\) on an F-rational (type) projective variety \(X\) of dimension \(d\) over a field, if the dimension of the complete linear system \(|L|\) associated to \(L\) exceeds \(d\), then \(K_X + nL\) is globally generated for all \(n \geq d\). A corollary is a special case of Fujita's conjecture. In characteristic zero this form of Fujita's conjecture can be straightforwardly proved by using the Kodaira vanishing theorem, but here the author gives a characteristic-free, purely algebraic proof by using tight closure theory, and avoiding the vanishing theorems completely. The key ingredient in Smith's proof is the behavior of the tight closure of the zero submodules in some local cohomology modules. For progress and survey on Fujita's conjecture, see \textit{U. Angehrn} and \textit{Y. T. Siu} [Invent. Math. 122, 291-308 (1995; Zbl 0847.32035)], \textit{L. Ein} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 6, 875-903 (1993; Zbl 0803.14004)], \textit{Y. Kawamata} [Math. Ann. 308, 491-505 (1997; Zbl 0909.14001)], \textit{J. Kollár} [in: Algebraic Geometry, Proc. Summer Res. Inst., Santa Cruz 1995, Proc. Symp. Pure Math. 62 (pt. 1), 221-287 (1997; Zbl 0905.14002)] and \textit{K. E. Smith} [J. Algebr. Geom. 6, 417-429 (1997; Zbl 0901.14005)].
scientific article; zbMATH DE number 1488548