| Budget Amount *help |
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Research Abstract |
1. Let E be an ample vector bundle of rank n-2【greater than or equal】 2 on a complex projective manifold X of dimension n having a section whose zero locus is a smooth surface Z. The structure of pairs (X, E) as above under the assumption that Z is a properly elliptic surface is determined. This generalizes known results on threefolds containing an elliptic surface as a smooth ample divisor.
2. Let E be a vector bundle of rank n - 1 on a smooth complex projective variety X of dimension n 【greater than or equal】 3, and let g(X, E) be the curve genus of (X, E) defined by the formula 2g(X, E) - 2 = (K_X + c_1(E))c_<n-1>(E), where K_X is the canonical bundle of X. Then it is proved that g(X, E) is a nonnegative integer if E is ample. Moreover, polarized pairs (X, E) with g(X, E) 【less than or equal】 1 are completely classified.
3. Let E be a very ample bundle of rank n - 1 on a smooth complex projective variety X of dimension n 【greater than or equal】 3, and let g(X, E) be the curve genus of (X, E) defined by the formula 2g(X, E) - 2 = (K_X + c_1(E))c_<n-1>(E), where K_X is the canonical bundle of X. Then the pairs (X, E) with g(X, E) = 2 are classified.
4. Let X be a smooth complex projective variety and let Z be a smooth submanifold of dimension 【greater than or equal】 2 of X, which is the zero locus of a section of an ample vector bundle E of rank dim X - dim Z 【greater than or equal】 2 on X. Let H be an ample line bundle on X whose restriction H_Z to Z is very ample. Triplets (X, E, H) as above are studied and classified under the assumption that Z is a projective manifold admitting a curve section which is a double cover of an elliptic curve.
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