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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus <i>g</i>(<i>d</i>,&#xa0;<i>s</i>) that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> can achieve assuming that it has degree <i>d</i> and does not lie on surfaces of degree less than <i>s</i>. In this paper, we prove that this prediction is correct when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(d=s\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(d\ge 2s-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> <mi>s</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We obtain this result by proving another conjecture, by Beorchia, Lella, and the second author, about initial ideals associated to certain homogeneous forms in a non-standard graded polynomial ring.</p>

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Initial ideals of weighted forms and the genus of locally Cohen-Macaulay curves

  • 期刊论文

  • 发表日期：

    2026年3月6日

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  • Alessio Sammartano,
  • Enrico Schlesinger

摘要

Let \(\mathcal {C}\) C be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus g(d, s) that \(\mathcal {C}\) C can achieve assuming that it has degree d and does not lie on surfaces of degree less than s. In this paper, we prove that this prediction is correct when \(d=s\) d = s or \(d\ge 2s-1\) d ≥ 2 s - 1 . We obtain this result by proving another conjecture, by Beorchia, Lella, and the second author, about initial ideals associated to certain homogeneous forms in a non-standard graded polynomial ring.