• 제목/요약/키워드: Minimal free resolution

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MINIMAL RESOLUTION CONJECTURES AND ITS APPLICATION

  • Cho, Young-Hyun
    • 대한수학회논문집
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    • 제13권2호
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    • pp.217-224
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    • 1998
  • In this paper we study the minimal resolution conjecture which is a generalization of the ideal generation conjecture. And we show how the results about this conjecture can make the calculation of minimal resolution in certain cases.

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THE MINIMAL FREE RESOLUTION OF THE UNION OF TWO LINEAR STAR-CONFIGURATIONS IN ℙ2

  • Shin, Yong-Su
    • 대한수학회논문집
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    • 제31권4호
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    • pp.683-693
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    • 2016
  • In [1], the authors proved that the finite union of linear star-configurations in $\mathbb{P}^2$ has a generic Hilbert function. In this paper, we find the minimal graded free resolution of the union of two linear star-configurations in $\mathbb{P}^2$ of type $s{\times}t$ with $\(^t_2\){\leq}s$ and $3{\leq}t$.

A GRADED MINIMAL FREE RESOLUTION OF THE 2ND ORDER SYMBOLIC POWER OF THE IDEAL OF A STAR CONFIGURATION IN ℙn

  • Shin, Yong-Su
    • 대한수학회지
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    • 제56권1호
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    • pp.169-181
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    • 2019
  • In [9], Geramita, Harbourne, and Migliore find a graded minimal free resolution of the 2nd order symbolic power of the ideal of a linear star configuration in ${\mathbb{P}}^n$ n of any codimension r. In [8], Geramita, Galetto, Shin, and Van Tuyl extend the result on a general star configuration in ${\mathbb{P}}^n$ but for codimension 2. In this paper, we find a graded minimal free resolution of the 2nd order symbolic power of the ideal of a general star configuration in ${\mathbb{P}}^n$ of any codimension r using a matroid configuration in [10]. This generalizes both the result on a linear star configuration in ${\mathbb{P}}^n$ of codimension r in [9] and the result on a general star configuration in ${\mathbb{P}}^n$ of codimension 2 in [8].

A GRADED MINIMAL FREE RESOLUTION OF THE m-TH ORDER SYMBOLIC POWER OF A STAR CONFIGURATION IN ℙn

  • Park, Jung Pil;Shin, Yong-Su
    • 대한수학회지
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    • 제58권2호
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    • pp.283-308
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    • 2021
  • In [30] the author finds a graded minimal free resolution of the 2-nd order symbolic power of a star configuration in ℙn of any codimension r. In this paper, we find that of any m-th order symbolic power of a star configuration in ℙn of codimension 2, which generalizes the result of Galetto, Geramita, Shin, and Van Tuyl in [15, Theorem 5.3]. Furthermore, we extend it to the m-th order symbolic power of a star configuration in ℙn of any codimension r for m = 3, 4, which also generalizes the result of Biermann et al. in [1, Corollaries 4.6 and 5.7]. We also suggest how to find a graded minimal free resolution of the m-th order symbolic power of a star configuration in ℙn of any codimension r for m ≥ 5.

CONSECUTIVE CANCELLATIONS IN FILTERED FREE RESOLUTIONS

  • Sharifan, Leila
    • 대한수학회보
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    • 제56권4호
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    • pp.1077-1097
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    • 2019
  • Let M be a finitely generated module over a regular local ring (R, n). We will fix an n-stable filtration for M and show that the minimal free resolution of M can be obtained from any filtered free resolution of M by zero and negative consecutive cancellations. This result is analogous to [10, Theorem 3.1] in the more general context of filtered free resolutions. Taking advantage of this generality, we will study resolutions obtained by the mapping cone technique and find a sufficient condition for the minimality of such resolutions. Next, we give another application in the graded setting. We show that for a monomial order ${\sigma}$, Betti numbers of I are obtained from those of $LT_{\sigma}(I)$ by so-called zero ${\sigma}$-consecutive cancellations. This provides a stronger version of the well-known cancellation "cancellation principle" between the resolution of a graded ideal and that of its leading term ideal, in terms of filtrations defined by monomial orders.

$\kappa$-CONFIGURATIONS IN $\mathbb{P}^2$ AND GORENSTEIN IDEALS OF CODIMENSION 3

  • Shin, Yong-Su
    • Journal of applied mathematics & informatics
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    • 제4권1호
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    • pp.249-261
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    • 1997
  • We find a necessary and sufficient condition for a $\kappa$-confi-guration $\mathbb{X}$ in $\mathbb{P}^2$ to be in generic position. We obtain the number and degrees of minimal generators of some Gorenstein ideals of codimension 3 and so obtain their minimal free resolution s of these ideals.

ON THE EQUATIONS DEFINING SOME RATIONAL CURVES OF MAXIMAL GENUS IN ℙ3

  • Wanseok LEE;Shuailing Yang
    • East Asian mathematical journal
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    • 제40권3호
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    • pp.287-293
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    • 2024
  • For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations and the syzygies among them. In this paper, we precisely determine a minimal generating set and the minimal free resolution of defining ideals of some rational curves of maximal genus in ℙ3.

ON THE MINIMAL FREE RESOLUTION OF CURVES OF MAXIMAL REGULARITY

  • Lee, Wanseok;Park, Euisung
    • 대한수학회보
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    • 제53권6호
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    • pp.1707-1714
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    • 2016
  • Let $C{\subset}{\mathbb{P}}^r$ be a nondegenerate projective curve of degree d > r + 1 and of maximal regularity. Such curves are always contained in the threefold scroll S(0, 0, r - 2). Also some of such curves are even contained in a rational normal surface scroll. In this paper we study the minimal free resolution of the homogeneous coordinate ring of C in the case where $d{\leq}2r-2$ and C is contained in a rational normal surface scroll. Our main result provides all the graded Betti numbers of C explicitly.

THE MINIMAL GRADED FREE RESOLUTION OF THE UNION OF TWO STAR CONFIGURATIONS IN 𝕡n AND THE WEAK LEFSCHETZ PROPERTY

  • Shin, Yong-Su
    • 충청수학회지
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    • 제30권4호
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    • pp.435-443
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    • 2017
  • We find a graded minimal free resolution of the union of two star configurations ${\mathbb{X}}$ and ${\mathbb{Y}}$ (not necessarily linear star configurations) in ${\mathbb{P}}^n$ of type s and t for s, $t{\geq}2$, and $n{\geq}3$. As an application, we prove that an Artinian ring $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ of two linear star configurations ${\mathbb{X}}$ and ${\mathbb{Y}}$ in ${\mathbb{P}}^3$ of type s and t has the weak Lefschetz property for $s{\geq}{\lfloor}\frac{1}{2}(^t_2){\rfloor}$ and $t{\geq}2$.