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• W2004661739 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G r left-parenthesis k comma n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Gr(k,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Plücker embedding of the Grassmann variety of projective <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-planes in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper P Superscript n> <mml:semantics> <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:annotation encoding=application/x-tex>mathbb P^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For a projective variety <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>, let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Subscript s Baseline left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma _s(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the variety of its secant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis s minus 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(s-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-planes. More precisely, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Subscript s Baseline left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma _s(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the Zariski closure of the union of linear spans of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-tuples of points lying on <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>. We exhibit two functions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s 0 left-parenthesis n right-parenthesis less-than-or-equal-to s 1 left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>s_0(n)le s_1(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Subscript s Baseline left-parenthesis upper G r left-parenthesis 2 comma n right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma _s(Gr(2,n))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the expected dimension whenever <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 9> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>9</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>ngeq 9</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and either <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s less-than-or-equal-to s 0 left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sle s_0(n)</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=s 1 left-parenthesis n right-parenthesis less-than-or-equal-to s> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>s_1(n)le s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Both <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s 0 left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>s_0(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s 1 left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>s_1(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are asymptotic to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartFraction n squared Over 18 EndFraction> <mml:semantics> <mml:mfrac> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mn>18</mml:mn> </mml:mfrac> <mml:annotation encoding=application/x-tex>frac {n^2}{18}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This yields, asymptotically, the typical rank of an element of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=logical-and Overscript 3 Endscripts double-struck upper C Superscript n plus 1> <mml:semantics> <mml:mrow> <mml:msup> <mml:mo movablelimits=false>⋀<!-- ⋀ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mspace width=thinmathspace /> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>bigwedge nolimits ^{3},{mathbb C}^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we classify all defective <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Subscript s Baseline left-parenthesis upper G r left-parenthesis k comma n right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma _s(Gr(k,n))</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=s less-than-or-equal-to 6> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>sle 6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and provide geometric arguments underlying each defective case." @default.
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• W2004661739 date "2011-01-03" @default.
• W2004661739 modified "2023-09-26" @default.
• W2004661739 title "Non-defectivity of Grassmannians of planes" @default.
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