• Korean Journal of Mathematics
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• 제6권2호
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• pp.291-298
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• 1998

The $L^p-L^q$ mapping properties of convolution operators with measures supported on curves in $\mathbb{R}^3$ have been studied by many authors. Oberlin provided examples of nondegenerate compact space curves whose arclength measures enjoy $L^p$-improving properties. This was later extended by Pan who showed that such properties hold for all nondegenerate compact space curves. In this paper, we will prove that the operator norm of the convolution operator with the arclength measure supported on a nondegenerate compact space curve depends only on certain quantities of the underlying curve.

• 대한수학회논문집
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• 제15권1호
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• pp.103-109
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• 2000

In this paper, we present the proof of generalized Fenchel's theorem by estimating the Gauss-Kronecker curvature of the tube of a nondegenerate closed curve in R$^{n}$ .

• East Asian mathematical journal
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• 제35권1호
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• pp.51-58
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• 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
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• 제34권1호
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• pp.21-28
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• 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 ${\mathbb{P}}^3$.

• East Asian mathematical journal
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• 제36권3호
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• pp.349-357
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• 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.

• East Asian mathematical journal
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• 제38권3호
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• pp.347-355
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• 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.

• 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.

• East Asian mathematical journal
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• 제40권1호
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• pp.75-83
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• 2024

For a reduced, irreducible and nondegenerate curve C ⊂ ℙ^r of degree d, it was shown that the arithmetic genus g of C has an upper bound π₀(d, r) by G. Castelnuovo. And he also classified the curves that attain the extremal value. These curves are arithmetically Cohen-Macaulay and contained in a surface of minimal degree. In this paper, we investigate the arithmetic genus of curves lie on a surface of minimal degree - the Veronese surface, smooth rational normal surface scrolls and singular rational normal surface scrolls. We also provide a construction of curves on singular rational normal surface scroll S(0, 2) ⊂ ℙ³ which attain the maximal arithmetic genus.