Classification of algebraic varieties with the chain condition for adjunction
In his paper in ''Algebraic varieties and analytic varieties'', Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 131-180 (1983), \textit{M. Reid} gave a definition for a minimal model of a projective threefold V as a 3- dimensional projective variety X with only terminal singularities which is birationally equivalent to V and such that either its canonical class \(K_ X\) is numerically effective, or the anticanonical class \(-K_ X\) is ample on fibers of some projective morphism \(f:X\to B\) with \(\dim B<\dim X.\) Reid conjectured that each threefold has a minimal model. The last results of Mori-Kawamata-Shokurov are very close to the complete proof of Reid's conjecture. - The author shows that if on a threefold V all the adjunction series \(| D+nK_ V|\) terminate for \(n\gg 0,\) then \(K_ X\) is not numerically effective for any model of V with canonical singularities. Hence, if V has a minimal model, then it is birationally equivalent to a variety of one of the following types: a Fano variety (possibly singular), a Del Pezzo fibering over some curve or a conic bundle over some surface. This is the Iskovskikh conjecture for threefolds with chain condition for adjunctions. Thus the Iskovskikh conjecture is proved modulo Reid's conjecture.
scientific article; zbMATH DE number 3863365