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<p>Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {P}^{\,n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mspace width="0.166667em" /> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra. This gives in particular new restrictions which must be satisfied by such graded Betti numbers. We define a homologically strictly plus-one generated hypersurface, and show that such a hypersurface has a singular locus of dimension <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n-2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> under some conditions.</p>

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On the degree of the singular subscheme of hypersurfaces in \(\mathbb {P}^{\,n}\)

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Alexandru Dimca,
  • Gabriel Sticlaru

摘要

Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in \(\mathbb {P}^{\,n}\) P n are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra. This gives in particular new restrictions which must be satisfied by such graded Betti numbers. We define a homologically strictly plus-one generated hypersurface, and show that such a hypersurface has a singular locus of dimension \(n-2\) n - 2 under some conditions.