TY - JOUR

T1 - On Seshadri constants of varieties with large fundamental group

AU - Di Cerbo, Gabriele

AU - Di Cerbo, Luca F.

N1 - Funding Information:
G. Di Cerbo is partially supported by the Simons Foundation. L. F. Di Cerbo is partially supported by a grant of the Max Planck Society: “Complex Hyperbolic Geometry and Toroidal Compactifications”, and by a grant associated to the S. S. Chern position at ICTP. Received September 19, 2016; accepted in revised form July 17, 2017. Published online February 2019.
Publisher Copyright:
© 2019 Scuola Normale Superiore. All rights reserved.

PY - 2019

Y1 - 2019

N2 - Let X be a smooth variety and let L be an ample line bundle on X. If πalg 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 π1(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-1B+(L) = B+(p∗L), where B+(L) is the augmented base locus of L.

AB - Let X be a smooth variety and let L be an ample line bundle on X. If πalg 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 π1(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-1B+(L) = B+(p∗L), where B+(L) is the augmented base locus of L.

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U2 - 10.2422/2036-2145.201609_012

DO - 10.2422/2036-2145.201609_012

M3 - Article

AN - SCOPUS:85068932227

SN - 0391-173X

VL - 19

SP - 335

EP - 344

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

JF - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

IS - 1

ER -