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<p>The purpose of this note is twofold. First, we give a quick proof of Ballico–Chiantini’s theorem stating that a Fano or Calabi–Yau variety of dimension at least 4 in codimension 2 is a complete intersection. Second, we improve Barth–Van de Ven’s result asserting that if the degree of a smooth projective variety of dimension <i>n</i> is less than approximately <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0.63 \cdot n^{1/2},\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.63</mn> <mo>·</mo> <msup> <mi>n</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> then it is a complete intersection. We show that the degree bound can be improved to approximately <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(0.79 \cdot n^{2/3}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.79</mn> <mo>·</mo> <msup> <mi>n</mi> <mrow> <mn>2</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Some remarks on smooth projective varieties of small degree and codimension

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  • 发表日期：

    2026年2月12日

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  • Jinhyung Park

摘要

The purpose of this note is twofold. First, we give a quick proof of Ballico–Chiantini’s theorem stating that a Fano or Calabi–Yau variety of dimension at least 4 in codimension 2 is a complete intersection. Second, we improve Barth–Van de Ven’s result asserting that if the degree of a smooth projective variety of dimension n is less than approximately \(0.63 \cdot n^{1/2},\) 0.63 · n 1 / 2 , then it is a complete intersection. We show that the degree bound can be improved to approximately \(0.79 \cdot n^{2/3}.\) 0.79 · n 2 / 3 .