• Journal of applied mathematics & informatics
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• 제11권1_2호
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• pp.437-444
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• 2003

A toric variety is defined by a certain collection of cones. Especially a toric Fano variety is obtained from a special nonsingular fan. In this paper, we define the f-vectors of toric Fano varieties as the numbers of faces of the corresponding fans, and investigate the f-vectors of some toric Fano varieties.

• 대한수학회지
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• 제59권1호
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• pp.87-103
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• 2022

We construct new families of smooth Fano fourfolds with Picard rank 1 which contain open 𝔸¹-cylinders, that is, Zariski open subsets of the form Z × 𝔸¹, where Z is a quasiprojective variety. In particular, we show that every Mukai fourfold of genus 8 is cylindrical and there exists a family of cylindrical Gushel-Mukai fourfolds.

• 대한수학회보
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• 제60권6호
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• pp.1705-1714
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• 2023

We explicitly construct the smooth toric Fano variety which is isomorphic to the blow-up of the projective space at torus invariant points in codimension one by anti-flips.

• 대한수학회보
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• 제54권5호
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• pp.1485-1526
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• 2017

We show that birational rigidity of Mori fibre spaces is not open in moduli.

• 대한수학회보
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• 제60권2호
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• pp.485-494
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• 2023

Let Y be the quintic del Pezzo 4-fold defined by the linear section of Gr(2, 5) by ℙ⁷. In this paper, we describe the locus of double lines in the Hilbert scheme of coincs in Y. As a corollary, we obtain the desigularized model of the moduli space of stable maps of degree 2 in Y. We also compute the intersection Poincaré polynomial of the stable map space.

• 대한수학회보
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• 제48권2호
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• pp.377-386
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• 2011

For a smooth algebraic curve C of genus g $\geq$ 4, let $SU_C$(r, d) be the moduli space of semistable bundles of rank r $\geq$ 2 over C with fixed determinant of degree d. When (r,d) = 1, it is known that $SU_C$(r, d) is a smooth Fano variety of Picard number 1, whose rational curves passing through a general point have degree $\geq$ r with respect to the ampl generator of Pic($SU_C$(r, d)). In this paper, we study the locus swept out by the rational curves on $SU_C$(r, d) of degree < r. As a by-product, we present another proof of Torelli theorem on $SU_C$(r, d).

• 대한수학회보
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• 제31권1호
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• pp.15-23
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• 1994

Throughout this paper, we are working on the complex number field C. The aim of this paper is to explain the applications of Theorem 2 in .cint. 1. In the surface theory, the adjoint linear system has played important roles and many tools have been developed to understand it. In the cases of higher dimensional varieties, we don't have any useful tools so far. Theorem 2 implies that it is enough to compute the dimension of the adjoint linear system to check the birationality. We can compute, somehow, the dimension of the adjoint linear system. For example, we can get an information about $h^{0}$ (X, $O_{x}$( $K_{x}$ + D)) from Euler characteristic of vertical bar $K_{X}$ + D vertical bar and some vanishing theorems. We are going to show the applications of Theorem 2 to smooth three-folds and smooth fourfold, specially, of general type with a nef canonical divisor, smooth Fano variety, and Calabi-Yau manifold. Our main results are Theorem A and Theorem B. Most of birationality problems in Theorem A and Theorem B have been studied. (see Ando [1] and Matsuki [4] for the detail matters.) But Theorem 2 gives short and easy proofs in the cases of dimension 3 and improves the previously known results in the cases of dimension 4.4. 4.4.