### Abstract

Let X be a smooth variety and let L be an ample line bundle on X. If π^{alg} _{1} (X) is large, we show that the Seshadri constant ϵ(p^{∗}L) can be made arbitrarily large by passing to a finite étale cover p : X^{'} → X. This result answers affirmatively to a conjecture of J.-M. Hwang. Moreover, we prove an analogous result when π_{1}(X) is large and residually finite. Finally, under the same topological assumptions, we appropriately generalize these results to the case of big and nef line bundles. More precisely, given a big and nef line bundle L on X and a positive number N > 0, we show that there exists a finite étale cover p : X^{'} → X such that the Seshadri constant ϵ(p^{∗}L; x) ≥ N for any x ϵ p^{-1}B_{+}(L) = B+(p^{∗}L), where B_{+}(L) is the augmented base locus of L.

Original language | English (US) |
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Pages (from-to) | 335-344 |

Number of pages | 10 |

Journal | Annali della Scuola Normale Superiore di Pisa - Classe di Scienze |

Volume | 19 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Mathematics (miscellaneous)

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## Cite this

*Annali della Scuola Normale Superiore di Pisa - Classe di Scienze*,

*19*(1), 335-344. https://doi.org/10.2422/2036-2145.201609_012