A bound on the degree of schemes defined by quadratic equations

Alzati, Alberto; Sierra, José Carlos

doi:10.1515/form.2011.081

Cited by 2 publications

(1 citation statement)

References 11 publications

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“…Remark 3.10. (Degree bound by property N 2,p ) Recently, A. Alzati and J.C. Sierra get a bound of quadrics for N 2,2 as paying attention to the structures of the rational map associated to the linear system of quadrics defining X, which coincides with our bound LB 2 (see [AS10]). They also derive a degree bound in terms of codimension e, d 2 ≤ 2e−1 e−1 whose asymptotic behavior is 2 e / 4 √ πe and describe the equality condition: this holds if and only if the equality of LB 2 holds.…”

Section: Embedded Linear Syzygies and Applicationssupporting

confidence: 65%

Analysis on some infinite modules, inner projection, and applications

Han

Kwak

2012

Trans. Amer. Math. Soc.

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A projective scheme $X$ is called `quadratic' if $X$ is scheme-theoretically cut out by homogeneous equations of degree 2. Furthermore, we say $X$ satisfies `property $\textbf{N}_{2,p}$' if it is quadratic and the quadratic ideal has only linear syzygies up to first $p$-th steps. In the present paper, we compare the linear syzygies of the inner projections with those of $X$ 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, \cite{EGHP1}. As an immediate corollary, we show that the inner projections of $X$ satisfy property $\textbf{N}_{2,p-1}$ for any reduced scheme $X$ with property $\textbf{N}_{2,p}$. Moreover, we also obtain the neccessary lower bound $(\codim X)\cdot p -\frac{p(p-1)}{2}$, which is sharp, on the number of quadrics vanishing on $X$ in order to satisfy $\textbf{N}_{2,p}$ and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme $X$. These results admit an interesting `syzygetic' rigidity theorem on property $\textbf{N}_{2,p}$ which leads the classifications of extremal and next to extremal cases. For these results we develope the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by M. Green. This new method allows us to treat a wider class of projective schemes which can not be covered by the Koszul cohomology techniques, because these are not projectively normal in general.Comment: 22 pages, minor changes (example 3.12 corrected, references updated, etc.), to appear in Trans. of Amer. Math. So

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Section: Embedded Linear Syzygies and Applicationssupporting

confidence: 65%

Analysis on some infinite modules, inner projection, and applications

Han

Kwak

2012

Trans. Amer. Math. Soc.

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The reduction number and degree bound of projective subschemes

Cường¹,

Kwak

2019

Trans. Amer. Math. Soc.

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In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen–Macaulay with linear resolutions. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give the possible explicit Betti tables for almost maximal cases. In addition, interesting examples are provided to understand our main results.

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A bound on the degree of schemes defined by quadratic equations

Cited by 2 publications

References 11 publications

Analysis on some infinite modules, inner projection, and applications

Analysis on some infinite modules, inner projection, and applications

The reduction number and degree bound of projective subschemes

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