• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.385-390
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• 1996

Let X be a smooth projective threefold whose anticanonical division $-K_X$ is ample, i.e., Fano threefold. In this paper, we studied the linear system $$\mid$-nK_X$\mid$$ for a positive integer n. In Theorem 4, we studied the cases that $\-nK_X$\mid$$ has no base-points and the cases that $$\mid$-nK_X$\mid$$ generate the birational map. In Proposition 5, we studied the possible exceptional cases given in Theorem 4. Some results in this paper are already known, but we have gave brief proofs for those results.

• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.357-375
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• 2024

The smooth quintic del Pezzo variety Y is well-known to be obtained as a linear sections of the Grassmannian variety Gr(2, 5) under the Plücker embedding into ℙ⁹. Through a local computation, we show the Hilbert scheme of conics in Y for dimY ≥ 3 can be obtained from a certain Grassmannian bundle by a single blowing-up/down transformation.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.425-432
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• 2013

Let S be a minimal surface of general type with $p_g(S)=q(S)=0$ having an involution ${\sigma}$ over the field of complex numbers. It is well known that if the bicanonical map ${\varphi}$ of S is composed with ${\sigma}$, then the minimal resolution W of the quotient $S/{\sigma}$ is rational or birational to an Enriques surface. In this paper we prove that the surface W of S with $K^2_S=5,6,7,8$ having an involution ${\sigma}$ with which the bicanonical map ${\varphi}$ of S is composed is rational. This result applies in part to surfaces S with $K^2_S=5$ for which ${\varphi}$ has degree 4 and is composed with an involution ${\sigma}$. Also we list the examples available in the literature for the given $K^2_S$ and the degree of ${\varphi}$.