# Classification Theory of Polarized Varieties by Takao Fujita

By Takao Fujita

Utilizing ideas from summary algebraic geometry which were constructed over fresh a long time, Professor Fujita develops category theories of such pairs utilizing invariants which are polarized higher-dimensional types of the genus of algebraic curves. the center of the booklet is the idea of D-genus and sectional genus constructed by way of the writer, yet various comparable subject matters are mentioned or surveyed. Proofs are given in complete within the critical a part of the advance, yet heritage and technical effects are often sketched in while the main points should not crucial for figuring out the main principles.

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Next we use 0 -+ OV , [ 2L - E] -+ 0 - OE -* 0. We have R1p*OV,[2L - E] = 0 for every fiber so 0 - OP[2H ] -+ 3 -+ Op - Hence Xx. 11). Then We have S c B V' * and is IHC - 2p HEI p: W -+ P. ) # 2 RB,F(W) V' S so that we can apply B E for some since f: x Xx -. W x is smooth along B = S + B1 is exact, 0 Note that H1(P, 0(2)) = 0. is the unique member of S = f(E) B HI(Xx, 2L - E) = 0 since since V' 12FI, F e Pic(W). is smooth along for some other component B1 with generated by H. [B1] = bH and p H .

4)) and that point on B. V Then is locally Cohen Macaulay at each (V, L) has a ladder. 2)]. V is assumed to be locally Macaulay everywhere, but it is easy to generalize as above. changed. If n = 2, nothing need be Indeed, if D n > 2, any general member of is a ILI rung by the argument in [F25]. 6), 2-Macaulay at any point not on B, so we get a ladder by induction. D is locally The details are left to the reader. §5. Classification of polarized varieties of d-genus zero We will classify all the polarized varieties of d-genus zero by the Apollonius method.

Is Such a fibration is not unique in general. 5: Birational classification of algebraic varieties the contraction morphism of this exteremal ray. R is called an extremal curve belonging to this ray. §5. 1) Definition (cf. [Iii], [1i2], [U], [F8]). be a line bundle on a variety if m > 0, and for any ImLI = 0 otherwise, where PImLI x = Maxm>0(dim PImLI(v)) is the rational map defined by ImLI. is sometimes denoted just by x(L, V) KV smooth and if x(KV, V) V. 2) Remark. V If x(L). is its canonical bundle, then is called the Kodaira dimension of x(V).