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Zariski decomposition is a very useful tool in order to study pseudo-effective
divisors on projective varieties. In particular we will investigate the semiampleness of the
positive part of the Zariski decomposition of big divisors on LC pairs whose difference
with the log-canonical divisor is nef. We will give a conjecture that would generalize an important
theorem by Kawamata and prove it up to dimension 3. In the second part of the talk we will focus on asymptotic base loci of line bundles: we will present a result, obtained in collaboration with Lorenzo Di Biagio, stating that the restricted base locus of every divisor on a KLT pair coincide with its non-nef locus, dened via numerical valuations. As a corollary we will see a characteriaztion of nef and abundant divisors involving asymptotic multiplier ideals that generalizes a theorem by F. Russo. |