• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.1067-1086
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• 2024

For a foliation 𝓕 of degree r over ℙ², we can regard it as a maximal invertible sheaf N^∨_𝓕 of Ω_ℙ², which is represented by a section s ∈ H⁰(Ω_ℙ² (r+2)). The singular locus Sing𝓕 of 𝓕 is the zero dimensional subscheme Z(s) of ℙ² defined by s. Campillo and Olivares have given some characterizations of the singular locus by using some cohomology groups. In this paper, we will give some different characterizations. For example, the singular locus of a foliation over ℙ² can be characterized as the residual subscheme of r collinear points in a complete intersection of two curves of degree r + 1.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.35-54
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• 2007

Let $M_c$ = M(2, 0, c) be the moduli space of O(l)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0\;and\;c_2=c$ on a K3 surface X, where O(1) is a generic ample line bundle on X. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, $M_c$ has trivial canonical divisor. In [22], O'Grady asks if there is any symplectic desingularization of $M_{2n}$ for $n\geq3$. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq3$. This obviously implies that there is no symplectic desingularization.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.325-335
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• 2018

We prove every oriented compact cyclic 3-orbifold has a contact structure. There is another proof in the web by Daniel Herr in his uploaded thesis which depends on open book decompositions, ours is independent of that. We define overtwisted contact structures, tight contact structures and Lutz twist on oriented compact cyclic 3-orbifolds. We show that every contact structure in an oriented compact cyclic 3-orbifold contactified by our method is homotopic to an overtwisted structure with the overtwisted disc intersecting the singular locus of the orbifolds. In course of proving the above results we prove a classification result for compact oriented cyclic-3 orbifolds which has not been seen by us in literature before.

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

• Journal of the Chungcheong Mathematical Society
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• v.37 no.2
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• pp.57-66
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• 2024

In this paper, we study how the Hilbert polynomial, associated with a reduced closed subscheme X of codimension 2 in ℙ^N, reveals geometric information about X. Although it is known that the Hilbert polynomial can tell us about the scheme's degree and arithmetic genus, we find additional geometric information it can provide for smooth varieties of codimension 2. To do this, we introduce the concept of Gotzmann coefficients, which helps to extract more information from the Hilbert polynomial. These coefficients are based on the binomial expansion of values of the Hilbert function. Our method involves combining techniques from initial ideals and partial elimination ideals in a novel way. We show how these coefficients can determine the degree of certain geometric features, such as the singular locus appearing in a generic projection, for smooth varieties of codimension 2.

• Journal of English Language & Literature
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• v.58 no.1
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• pp.43-67
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• 2012

My essay aims at illustrating Whitman's homosexual vision of utopia with a close reading of his representative homosexual text, Calamus. His expansive self is based upon his intimate contact with the world and is almost always drawn to a wider vision of community in which different individuals share the locus of commonness and reach beyond their empirical boundaries. While foregrounding the contingent and the singular, Whitman forges bonds with other people through a series of ecstatic moments that carry us into the public sphere and common interests. Contrary to the current Whitman studies, his homosexual text doesn't repress contingency in order to celebrate the universal, but fully develops the commensurability among diverse historical agents. Whitman knows well the social taboos and inhibitions at the time of national crisis and expansion, but keeps imagining the world where homosexuality plays a central and significant role in founding a democratic solidarity and achieving a desirable social structure. His ideal of America is not a deferred wish for the future, but a concrete vision that can be achieved here and now, realized by the spontaneous bonding and instant attraction among free men. Instead of interpreting history or suggesting practical alternatives, he keeps questioning the dominant ideologies and the given orders of social control, and suggests a free and open relationship among men where no exterior power or mediating other intervenes. His utopian vision is radical as well as ideal, in that it rejects the interventions of the power structure and its institutions and courageously inscribes his homosexuality in the process of writing about and reading his contemporary America. As a predecessor of a homosexual utopian vision of America, Whitman has inspired many later poets, showing a possibility of infusing a homosexual identity into a radical imaging of the nation and its future.