# Fujita-type conjectures and Seshadri constants

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Takumi Murayama, University of Michigan
Fine Hall 322

If L is an ample divisor on a projective variety X, then K+mL is basepoint-free for large enough m, where K is the canonical divisor. T.

Fujita in 1988 conjectured that if X is n-dimensional, then m = n+1 should suffice for basepoint-freeness. To study this conjecture, Demailly introduced Seshadri constants as a way to measure the local positivity of L. While examples of Miranda show that Seshadri constants cannot answer Fujita's conjecture, Seshadri constants can still give information about basepoint-freeness at general points. We will present joint work with Yajnaseni Dutta, in which we exploit Seshadri constants to give positive evidence toward Popa and Schnell's relative version of Fujita's conjecture, which asks a similar question for pushforwards of log-pluricanonical sheaves.