• 대한수학회논문집
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• 제18권4호
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• pp.603-613
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• 2003

When X is a threefold of general type, it is well known h/sup 0/(X, O/sub X/(nK/sub X/)) ≥ 1 for a sufficiently large n. When X(O/sub X/) 〉 0, it is not easy to obtain such an integer n. A. R. Fletcher showed that h/sup 0/(X, O/sub X/(nK/sub X/)) ≥ 1 for n = 12 when X(O/sub X/)=1. We introduce a technique different from A. R. Fletcher's. Using this technique, we also prove the same result as he showed and have a new result.

• 대한수학회논문집
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• 제25권2호
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• pp.161-165
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• 2010

Even though there is a formula for $h^0$(X, $\cal{O}_X(nK_X)$) for a canonical threefold X, it is not easy to compute $h^0$(X, $\cal{O}_X(nK_X)$) because the formula has a term due to singularities. In this paper, we find a way to control the term due to singularities. We show nonvanishing of plurigenus for the case when the index r in the singularity type $\frac{1}{r}$(1, -1, b) is sufficiently large.

• 대한수학회논문집
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• 제27권1호
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• pp.37-46
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• 2012

It is well known that plurigenus does not vanish for a sufficiently large multiple on a canonical threefold over $\mathbb{C}$. There is Reid Fletcher formula for plurigenus. But, unlike in the case of surface of general type, it is not easy to compute plurigenus. In this paper, we in-duce a different version of Reid-Fletcher formula and show that the constant term in the induced formula has periodic properties. Using these properties we have an application to nonvanishing of plurigenus.

• 대한수학회보
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• 제53권1호
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• pp.303-323
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• 2016

For a canonical threefold X, it is known that $p_n$ does not vanish for a sufficiently large n, where $p_n=h^0(X,\mathcal{O}_X(nK_X))$. We have shown that $p_n$ does not vanish for at least one n in {6, 8, 10}. Assuming an additional condition $p_2{\geq}1$ or $p_3{\geq}1$, we have shown that $p_{12}{\geq}2$ and $p_n{\geq}2$ for $n{\geq}14$ with one possible exceptional case. We have also found some inequalities between ${\chi}(\mathcal{O}_X)$ and $K^3_X$.

• 건축역사연구
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• 제13권2호
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• pp.21-38
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• 2004

This study provides a descriptive and analytical account of major aspects of urban development and transformation of housing types of Italian Firenze from the 13th century to the 19th century. It is a typo-morphological depiction of urban spatial structure of the extraordinary city, Firenze, the center of Italian Renaissance. And this study has proceeded on the assumption that the evolving form of the urban structure and housing types cannot be understood without reference to the larger context of political, economic, and social life. Based on these backgrounds, the purpose of this study is threefold: to provide a comprehensive discussion of general characteristics of urban spatial structure of Firenze, and to explain the process of formation of working-class neighborhoods by constructing new city wall in later 13th century, and to discuss transformation of housing types of the working-class neighborhood with understanding the mechanism of existence of housing in the newly formed residential neighborhoods. The development of residential neighborhoods was pursued by 'planned' manner through forming square-shaped blocks, and characterized by the subdivision of larger properties into standardized building lots for the construction of houses. On the bases of documentary evidences, several ecclesiastical institutions are identified as the agents of a distinctive type of development. While the institutions did the major role for developing lands, the construction of houses was done by small scale construction agents with moderate amount of properties. The major housing type of working-class neighborhoods of Firenze has been the 'casa a schiera' characterized by the form of narrow front and long depth. The type was generalized by the newly formed middle and working-class of Firenze which grew their body very rapidly, Even though the type assumed very uniform in its fen there were many variations. And through passing time, the casa a schiera developed to be multi-family housing, and the level of variation became deepen. Eventually, transformation of housing type of Firenze was ended by appearance of the 'casa in linea', which was very similar to modern apartment in its spatial organization.