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• W1645804778 abstract "We prove the following: <bold>Theorem.</bold> <italic>Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X subset-of bold upper P Superscript 5> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>P</mml:mi> </mml:mrow> <mml:mn>5</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>Xsubset mathbf {P}^5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth, subcanonical threefold. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h Superscript 0 Baseline left-parenthesis script upper I Subscript upper X Baseline left-parenthesis 4 right-parenthesis right-parenthesis not-equals 0> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>I</mml:mi> </mml:mrow> <mml:mi>X</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>h^0(mathcal {I}_X(4))ne 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a complete intersection.</italic> <italic>Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X subset-of bold upper P Superscript 6> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>P</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>Xsubset mathbf {P}^6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth, codimension two subvariety, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h Superscript 0 Baseline left-parenthesis script upper I Subscript upper X Baseline left-parenthesis 5 right-parenthesis right-parenthesis not-equals 0> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>I</mml:mi> </mml:mrow> <mml:msub> <mml:mspace width=negativethinmathspace /> <mml:mi>X</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>h^0(mathcal {I}!_X(5))!ne 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=degree left-parenthesis upper X right-parenthesis less-than-or-equal-to 73> <mml:semantics> <mml:mrow> <mml:mi>deg</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>73</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {deg}(X)le 73</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a complete intersection.</italic> This improves, for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=5 less-than-or-equal-to n less-than-or-equal-to 6> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>n</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>5le nle 6</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, earlier results on Hartshorne’s conjecture for codimension two subvarieties of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper P Superscript n> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbf {P}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1645804778 created "2016-06-24" @default.
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• W1645804778 date "2002-03-21" @default.
• W1645804778 modified "2023-09-24" @default.
• W1645804778 title "On codimension two subvarieties of 𝑃⁵ and 𝑃⁶" @default.
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