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March 2018 A Note on Brill–Noether Existence for Graphs of Low Genus
Stanislav Atanasov, Dhruv Ranganathan
Michigan Math. J. 67(1): 175-198 (March 2018). DOI: 10.1307/mmj/1519095622

Abstract

In an influential 2008 paper, Baker proposed a number of conjectures relating the Brill–Noether theory of algebraic curves with a divisor theory on finite graphs. In this note, we examine Baker’s Brill–Noether existence conjecture for special divisors. For g5 and ρ(g,r,d) nonnegative, every graph of genus g is shown to admit a divisor of rank r and degree at most d. As further evidence, the conjecture is shown to hold in rank 1 for a number families of highly connected combinatorial types of graphs. In the relevant genera, our arguments give the first combinatorial proof of the Brill–Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker.

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Stanislav Atanasov. Dhruv Ranganathan. "A Note on Brill–Noether Existence for Graphs of Low Genus." Michigan Math. J. 67 (1) 175 - 198, March 2018. https://doi.org/10.1307/mmj/1519095622

Information

Received: 12 September 2016; Revised: 3 November 2016; Published: March 2018
First available in Project Euclid: 20 February 2018

zbMATH: 06965595
MathSciNet: MR3770859
Digital Object Identifier: 10.1307/mmj/1519095622

Subjects:
Primary: 14H51, 14T05

Rights: Copyright © 2018 The University of Michigan

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Vol.67 • No. 1 • March 2018
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