The Barth quintic surface has Picard number 41

• Slawomir Rams ^[1] ; Matthias Schütt ^[2]
  1. [1] Jagiellonian University

     Jagiellonian University

     Kraków, Polonia

     
  2. [2] University of Hannover

     University of Hannover

     Region Hannover, Alemania

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 13, Nº 2, 2014, págs. 533-549
• Idioma: inglés
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• Resumen

  • This paper investigates a specific smooth quintic surface suggested by Barth for it contains the current record of 75 lines over the complex numbers. Our main incentive is to prove that the complex quintic has Picard number 41, and to compute the N´eron-Severi group up to a 2-power index. We also compute Picard numbers for reductions to positive characteristic and verify the Tate conjecture.