A tight closure proof of Fujita's freeness conjecture for very ample line bundles

Abstract

The proof here actually proves a stronger statement than Theorem 1 above. The variety X need not be smooth; F-rationality is sufficient (the definition is recalled in the next section; in characteristic zero it is equivalent to rational singularities). Also, the line bundle L need not be very ample; it is sufficient if L is globally generated and the dimension the complete linear system |L| is greater than d. These generalizations are summarized in Theorem 2. Furthermore, with a little more work, the same ideas prove even stronger statements, which are interesting algebraically, but difficult to interpret geometrically (see Theorem 3). Fujita’s Freeness Conjecture predicts the same conclusion under the much weaker hypothesis that L is only ample. While open in general, for varieties defined over a field of characteristic zero, it is known in dimension four or less [R], [EL], [Ka]. See also [AS] for important progress on the conjecture, and [Ko] for a good survey about it. In characteristic zero, it is not hard to give a geometric argument of the special case above using the Kodaira Vanishing Theorem. The goal here is to give a simple, quite different proof that is purely algebraic and valid in any characteristic. This argument offers a nice illustration of how tight closure can be used to prove geometric theorems in arbitrary characteristic without the use of the usual tools of desingularization or vanishing theorems.