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• W2046209512 abstract "In this paper, we prove that if<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X subset-of double-struck upper P Superscript n><mml:semantics><mml:mrow><mml:mi>X</mml:mi><mml:mo>⊂<!-- ⊂ --></mml:mo><mml:msup><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>P</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>Xsubset mathbb {P}^n</mml:annotation></mml:semantics></mml:math></inline-formula>,<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 4><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥<!-- ≥ --></mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>nge 4</mml:annotation></mml:semantics></mml:math></inline-formula>, is a locally complete intersection of pure codimension<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2><mml:semantics><mml:mn>2</mml:mn><mml:annotation encoding=application/x-tex>2</mml:annotation></mml:semantics></mml:math></inline-formula>and defined scheme-theoretically by three hypersurfaces of degrees<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d 1 greater-than-or-equal-to d 2 greater-than-or-equal-to d 3><mml:semantics><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>≥<!-- ≥ --></mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>≥<!-- ≥ --></mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>d_1ge d_2ge d_3</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 H Superscript 1 Baseline left-parenthesis double-struck upper P Superscript n Baseline comma script upper I Subscript upper X Baseline left-parenthesis j right-parenthesis right-parenthesis equals 0><mml:semantics><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:msup><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>P</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>,</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:mi>j</mml:mi><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^1(mathbb {P}^n,mathcal {I}_X(j))=0</mml:annotation></mml:semantics></mml:math></inline-formula>for<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=j greater-than d 3><mml:semantics><mml:mrow><mml:mi>j</mml:mi><mml:mo>></mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>j>d_3</mml:annotation></mml:semantics></mml:math></inline-formula>using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X subset-of double-struck 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=double-struck>P</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>Xsubset mathbb {P}^5</mml:annotation></mml:semantics></mml:math></inline-formula>is projectively normal if<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 defined by three quintic hypersurfaces." @default.
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• W2046209512 date "2005-12-14" @default.
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• W2046209512 title "On quasi-complete intersections of codimension 2" @default.
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