2008/07/31 by Kangjin Han, Sijong Kwak
Computer Science · Mathematics · #Algebra over a field #Algorithm #Commutative Algebra and Its Applications #Mathematics #Polynomial and algebraic computation #Projection (relational algebra) #Pure mathematics #Rings, Modules, and Algebras #math.AC #math.AG #msc:13D02 #msc:14N05 #msc:14N25 #msc:51N35
paper · pdf · doi:10.1090/s0002-9947-2012-05755-2
22 pages, minor changes (example 3.12 corrected, references updated, etc.), to appear in Trans. of Amer. Math. Soc
arxiv created 2012/05/15 · openalex publication_date 2012/06/08 · arxiv updated 2012/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A projective scheme <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 called ‘quadratic’ 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 scheme-theoretically cut out by homogeneous equations of degree <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> . Furthermore, we say that <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> satisfies ‘property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper N Subscript 2 comma p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">N</mml:mtext> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\textbf N2,p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ’ if it is quadratic and the quadratic ideal has only linear syzygies up to the first <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -th steps. In the present paper, we compare the linear syzygies of the inner projections with those of <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> and obtain a theorem on ‘embedded linear syzygies’ as one of our main results. This is the natural projection-analogue of ‘restricting linear syzygies’ in the linear section case. As an immediate corollary, we show that the inner projections of <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> satisfy property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper N Subscript 2 comma p minus 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">N</mml:mtext> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\textbf N2,p-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any reduced scheme <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> with property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper N Subscript 2 comma p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">N</mml:mtext> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\textbf N2,p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Moreover, we also obtain the neccessary lower bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis c o d i m upper X right-parenthesis dot p minus StartFraction p left-parenthesis p minus 1 right-parenthesis Over 2 EndFraction"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>codim</mml:mi> <mml:mo> </mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">(\operatorname codim X)⋅ p -\frac p(p-1)2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which is sharp, on the number of quadrics vanishing 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>