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In 1988, Fujita conjectured that there is an effective and uniform way toturn an ample line bundle on a smooth projective variety into a globallygenerated or very ample line bundle. We study Fujita's conjecture usingSeshadri constants, which were first introduced by Demailly in 1992 with thehope that they could be used to prove cases of Fujita's conjecture. Whileexamples of Miranda seemed to indicate that Seshadri constants could not beused to prove Fujita's conjecture, we present a new approach to Fujita'sconjecture using Seshadri constants and positive characteristic methods. Ourtechnique recovers some known results toward Fujita's conjecture over thecomplex numbers, without the use of vanishing theorems, and proves new resultsfor complex varieties with singularities. Instead of vanishing theorems, we usepositive characteristic techniques related to the Frobenius-Seshadri constantsintroduced by Musta\c{t}\u{a}-Schwede and the author. As an application of ourresults, we give a characterization of projective space using Seshadriconstants in positive characteristic, which was proved in characteristic zeroby Bauer and Szemberg.

Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods