Nakamaye's theorem on log canonical pairs

We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension at most 1. We also generalize Ein-Lazarsfeld-Mustata-Nakamaye-Popa's description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities.Comment: v2: We removed, in the introduction, the phrase about Choi's papers, as he uses Nakamaye's theorem in the semiample case. Updated references. v3: added reference to Ambro's "Quasi-log varieties". v4: improved exposition in sections 1, 2 and 4; slightly corrected the statement of Lemma 3.

Brill-Noether theory of curves on Enriques surfaces, II. The Clifford index

We complete our study of linear series on curves lying on an Enriques surface by showing that, with the exception of smooth plane quintics, there are no exceptional curves on Enriques surfaces, that is, curves for which the Clifford index is not computed by a pencil

Augmented base loci and restricted volumes on normal varieties

We extend to normal projective varieties defined over an arbitrary algebraically closed field a result of Ein, Lazarsfeld, Musta\c{t}\u{a}, Nakamaye and Popa characterizing the augmented base locus (aka non-ample locus) of a line bundle on a smooth projective complex variety as the union of subvarieties on which the restricted volume vanishes. We also give a proof of the folklore fact that the complement of the augmented base locus is the largest open subset on which the Kodaira map defined by large and divisible multiples of the line bundle is an isomorphism.Comment: 7 pages. v2: we made a small modification of the statement of Lemma 2.4, a few minor corrections and updated reference