• Title/Summary/Keyword: minimal free resolution

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

  • Park, Jung Pil;Shin, Yong-Su
    • Journal of the Korean Mathematical Society
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    • v.58 no.2
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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.

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

  • Shin, Yong-Su
    • Journal of the Korean Mathematical Society
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    • v.56 no.1
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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].

ON THE MINIMAL FREE RESOLUTION OF CURVES OF MAXIMAL REGULARITY

  • Lee, Wanseok;Park, Euisung
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.6
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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 FREE RESOLUTION OF CERTAIN DETERMINANTAL IDEA

  • CHOI, EUN-J.;KIM, YOUNG-H.;KO, HYOUNG-J.;WON, SEOUNG-J.
    • Communications of the Korean Mathematical Society
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    • v.20 no.2
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    • pp.275-290
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    • 2005
  • Let $S\;=\;R[\chi_{ij}\mid1\;{\le}\;i\;{\le}\;m,\;1\;{\le}\;j\;{\le}\;n]$ be the polynomial ring over a noetherian commutative ring R and $I_p$ be the determinantal ideal generated by the $p\;\times\;p$ minors of the generic matrix $(\chi_{ij})(1{\le}P{\le}min(m,n))$. We describe a minimal free resolution of $S/I_{p}$, in the case m = n = p + 2 over $\mathbb{Z}$.

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

  • Shin, Yong-Su
    • Journal of applied mathematics & informatics
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    • v.4 no.1
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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.

NEW CONSTRUCTION OF THE EAGON-NORTHCOTT COMPLEX

  • Kang, Oh-Jin;Kim, Joohyung
    • Korean Journal of Mathematics
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    • v.20 no.2
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    • pp.161-176
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    • 2012
  • The authors [6] introduced the concept of a complete matrix of grade $g$ > 3 to describe a structure theorem for complete intersections of grade $g$ > 3. We show that a complete matrix can be used to construct the Eagon-Northcott complex [7]. Moreover, we prove that it is the minimal free resolution $\mathbb{F}$ of a class of determinantal ideals of $n{\times}(n+2)$ matrices $X=(x_{ij})$ such that entries of each row of $X=(x_{ij})$ form a regular sequence and the second differential map of $\mathbb{F}$ is a matrix $f$ defined by the complete matrices of grade $n+2$.

CONSECUTIVE CANCELLATIONS IN FILTERED FREE RESOLUTIONS

  • Sharifan, Leila
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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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.

THE MINIMAL FREE RESOLUTION OF A STAR-CONFIGURATION IN ?n AND THE WEAK LEFSCHETZ PROPERTY

  • Ahn, Jea-Man;Shin, Yong-Su
    • Journal of the Korean Mathematical Society
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    • v.49 no.2
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    • pp.405-417
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    • 2012
  • We find the Hilbert function and the minimal free resolution of a star-configuration in $\mathbb{P}^n$. The conditions are provided under which the Hilbert function of a star-configuration in $\mathbb{P}^2$ is generic or non-generic We also prove that if $\mathbb{X}$ and $\mathbb{Y}$ are linear star-configurations in $\mathbb{P}^2$ of types t and s, respectively, with $s{\geq}t{\geq}3$, then the Artinian k-algebra $R/(I_{\mathbb{X}}+I_{\mathbb{Y})$ has the weak Lefschetz property.

SOME GEOMETRIC CONSEQUENCES OBTAINED FROM PARTIAL ELIMINATION IDEALS

  • Ahn, Jeaman
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.3
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    • pp.487-494
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    • 2010
  • In [9], M. Green introduced the partial elimination ideals defining the multiple loci of the projection image of a closed subscheme in ${\mathbb{P}}^n$. In this paper, we give some geometric consequences obtained from partial elimination ideals.