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Noether inequality on a threefolds with one-dimensional canonical image

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Shin, D.-K. Osaka J. Math. 41 (2004), 81–84 NOETHER INEQUALITY ON A THREEFOLDS WITH ONE-DIMENSIONAL CANONICAL IMAGE DONG-KWAN SHIN (Received July 22, 2002) Throughout this paper, we are working over the complex number field C. On a projective minimal surface of general type, Noether inequality ( ) = 0( O ( )) ≤ 1 2 2 + 2 holds where is the canonical divisor. Noether inequality made an important contri- bution to understanding the geography of surfaces of general type. Unfortunately, we can not extend Noether inequality to a threefold of general type. A threefold version of Noether inequality is the following. ( ) = 0( O ( )) ≤ 1 2 3 + dim Im | | where is a minimal threefold of general type. This version of Noether inequal- ity holds true when dim Im | | = 3. But when dim Im | | = 2, M. Kobayashi showed the existence of a counter example in M. Kobayashi [3, Proposition (3.2)]. When dim Im | | = 1, M. Kobayashi described the possible exceptional cases as- suming that is factorial. When dim Im | | = 1, we have the following: (0) ( ) ≤ (1/2) 3 + 1 or if not, we have the following two possible exceptional cases (1) is singular, the image is a rational curve, all the fibers are connected, 3 = 1 and ( ) = 2 (2) The map | | is a morphism and a general fiber is a normal algebraic irre- ducible surface with only canonical singularities which have ample canonical divisor, 2 = 1, ( ) = 0 and ( ) = 1 or 2, where ( ) and ( ) are the irregularity and the genus of respectively. For detail matters, see M. Kobayashi [3]. But the existence of each possible exceptional case — the case (1) or the case (2) — he described is not known yet. However, in the case (1), we have the additional information about the genus and 3 . In the case (2), we don’t have any such information. Thus, we need an addi- 2000 Mathematics Subject Classification : 14E05, 14J30. This paper was supported by KOSEF R01-1999-00004. 82 D.-K. SHIN tional information about the invariants of to describe in detail. I

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