Brill–Noether existence on graphs via R-divisors, polytopes and lattices

• Madhusudan Manjunath ^[1]
  1. [1] Indian Institute of Technology Bombay, India
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 2, 2022
• Idioma: inglés
• DOI: 10.1007/s00029-021-00728-0
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• Resumen

  • We study Brill–Noether existence on a finite graph using methods from polyhedral geometry and lattices. We start by formulating analogues of the Brill–Noether conjectures (both the existence and non-existence parts) for R-divisors, i.e. divisors with real coefficients, on a graph. We then reformulate the Brill–Noether existence conjecture for R-divisors on a graph in geometric terms, that we refer to as the covering radius conjecture and we show a weak version, in support of it. Using this, we show an approximate version of the Brill–Noether existence conjecture for divisors on a graph. As applications, we derive upper bounds on the gonality of a graph and its R-divisor analogue.