• Title, Summary, Keyword: Fano variety

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LOCI OF RATIONAL CURVES OF SMALL DEGREE ON THE MODULI SPACE OF VECTOR BUNDLES

  • Choe, In-Song
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.2
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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).

ON THE ADJOINT LINEAR SYSTEM

  • Kwan, Shin-Dong
    • Bulletin of the Korean Mathematical Society
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    • v.31 no.1
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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.

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