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<p>Let <i>G</i> be a finite graph of genus <i>g</i>. Let <i>d</i> and <i>r</i> be non-negative integers such that the Brill–Noether number is non-negative. Then the Brill–Noether existence conjecture due to Baker predicts the existence of a divisor of degree <i>d</i> and rank at least <i>r</i> on <i>G</i>.</p><p>The conjecture is known to be true on the <i>k</i>-th homothetic refinement <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(G^{(k)}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> of <i>G</i>, for <i>k</i> sufficiently large. Here we use results from classical Brill–Noether theory to give a uniform upper bound for <i>k</i> in terms of <i>g</i>,&#xa0;<i>d</i>, and <i>r</i>. We also discuss some algebro-geometric aspects of the conjecture and of recent counterexamples to a related conjecture found in van Dobben de Bruyn et al. (J Combin Theory Ser A 189:105619, 2022).</p>

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Bounding the number of graph refinements for Brill–Noether existence

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Karl Christ,
  • Qixiao Ma

摘要

Let G be a finite graph of genus g. Let d and r be non-negative integers such that the Brill–Noether number is non-negative. Then the Brill–Noether existence conjecture due to Baker predicts the existence of a divisor of degree d and rank at least r on G.

The conjecture is known to be true on the k-th homothetic refinement \(G^{(k)}\) G ( k ) of G, for k sufficiently large. Here we use results from classical Brill–Noether theory to give a uniform upper bound for k in terms of g, d, and r. We also discuss some algebro-geometric aspects of the conjecture and of recent counterexamples to a related conjecture found in van Dobben de Bruyn et al. (J Combin Theory Ser A 189:105619, 2022).