• Bulletin of the Korean Mathematical Society
• /
• v.55 no.6
• /
• pp.1749-1754
• /
• 2018

If a graph G is both claw-free and gap-free, then E. Nevo showed that the Castelnuovo-Mumford regularity of the associated edge ideal I(G) is at most three. Later Dao, Huneke and Schwieg gave a simpler proof of this result. In this paper we introduce a class of edge ideals with Castelnuovo-Munmford regularity at most four.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.4
• /
• pp.777-787
• /
• 2009

In this paper we investigate the upper bound on the Castelnuovo-Mumford regularity of a graded module with linear free presentation. Let M be a finitely generated graded module over a polynomial ring R with zero dimensional support. We prove that if M is generated by elements of degree $d{\geq}0$ with a linear free presentation $$\bigoplus^p{R}(-d-1)\longrightarrow^{\phi}\bigoplus^q{R}(-d){\longrightarrow}M{\longrightarrow}0$$, then the Castelnuovo-Mumford regularity of M is at most d+q-1. As an important application, we can prove vector bundle technique, which was used in [11], [13], [17] as a tool for obtaining several remarkable results.

• Communications of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.113-125
• /
• 2019

In this paper, we introduce the elimination ideal $I_D(G)$ associated to a simple finite graph G. We obtain the upper bound of Castelnuovo-Mumford regularity of elimination ideal for various classes of graphs.

• East Asian mathematical journal
• /
• v.36 no.3
• /
• pp.349-357
• /
• 2020

For a nondegenerate projective curve C ⊂ ℙ^r of degree d, it was shown that the Castelnuovo-Mumford regularity reg(C) of C is at most d - r + 2. And the curves of maximal regularity which attain the maximally possible value d - r + 2 are completely classified. In this short note, we first collect several known results about curves of maximal regularity. We provide a new proof and some partial results. Finally we suggest some interesting questions.

• Journal of Integrative Natural Science
• /
• v.11 no.2
• /
• pp.82-86
• /
• 2018

We show that the (Castelnuovo-Mumford) regularity of a homogeneous Cohen-Macaulay ring agrees with some of the invariants defined the papers, and we study a ring of small index.

• East Asian mathematical journal
• /
• v.35 no.1
• /
• pp.51-58
• /
• 2019

For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations. In this paper we precisely determine the defining equations of some rational curves of maximal regularity in ${\mathbb{P}}^4$ according to their rational parameterizations.

• East Asian mathematical journal
• /
• v.38 no.3
• /
• pp.347-355
• /
• 2022

For a nondegenerate projective variety, it is a classical problem to study its defining equations with respect to a given embedding. In this paper, we precisely determine minimal sets of generators of the defining ideals of some curves of maximal regularity in ℙ⁵.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.3
• /
• pp.487-494
• /
• 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.

• East Asian mathematical journal
• /
• v.34 no.1
• /
• pp.21-28
• /
• 2018

For a nondegenerate irreducible projective variety, it is a classical problem to study the defining equations of a variety with respect to the given embedding. In this paper we precisely determine the defining equations of certain types of rational curves in ${\mathbb{P}}^3$.

• Journal of the Chungcheong Mathematical Society
• /
• v.34 no.2
• /
• pp.181-188
• /
• 2021

Let X be a nondegenerate reduced closed subscheme in ℙⁿ. Assume that π_q : X → Y = π_q(X) ⊂ ℙ^n-1 is a generic projection from the center q ∈ Sec(X) \ X where Sec(X) = ℙⁿ. Let Z be the singular locus of the projection π_q(X) ⊂ ℙ^n-1. Suppose that I_X has the almost minimal presentation, which is of the form R(-3)^β_2,1 ⊕ R(-4) → R(-2)^β_1,1 → I_X → 0. In this paper, we prove the followings: (a) Z is either a linear space or a quadric hypersurface in a linear subspace; (b) $H^1({\mathcal{I}_X(k)})=H^1({\mathcal{I}_Y(k)})$ for all k ∈ ℤ; (c) reg(Y) ≤ max{reg(X), 4}; (d) Y is cut out by at most quartic hypersurfaces.