By Takao Fujita

Utilizing strategies from summary algebraic geometry which were built over fresh a long time, Professor Fujita develops category theories of such pairs utilizing invariants which are polarized higher-dimensional models of the genus of algebraic curves. the guts of the booklet is the idea of D-genus and sectional genus constructed via the writer, yet quite a few comparable issues are mentioned or surveyed. Proofs are given in complete within the critical a part of the improvement, yet history and technical effects are often sketched in while the main points aren't crucial for knowing the foremost principles.

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**Extra resources for Classification Theory of Polarized Varieties**

**Example text**

Is the restriction map r In particular and the equality holds if and only if such a case tivity of 4(D, LD) t d(V, L) is surjective. r is said to be a regular rung. D In The surjec- is important from rt: H0(V, tL) -+ HO(D, tLD) the ring theoretic viewpoint. 3) Theorem (cf. 1)] & [F2;Prop. 2]). be a polarized variety and let (V, L) JaLl defined by D be a member of for some 6 E H0(V, aL) Let a > 0. Let. 1, k be homogeneous elements of the graded algebra G(V, L) = ED tk0H0(V, tL) images in G(D, LD) is generated by Proof.

Let. 1, k be homogeneous elements of the graded algebra G(V, L) = ED tk0H0(V, tL) images in G(D, LD) is generated by Proof. Let generated by Then the 8 A 8 and the j's. Set Then G(V, L) At = A n H0(V, tL). G(D, LD) Hence it suffices to show exact sequence Suppose that j's. be the subalgebra of and the be their nj's as an algebra. rt(At) = HO(D, tLD), for r/j's. nk , nl' via the restriction. is generated by the G(D, LD) G(V, L) and let is generated by Ker(rt) C At. 4) A line bundle t. is said to be simply generated L In this case the rational map H0(V, L).

If is V char(s) = 0, this is proved by using Mori- smooth and Kawamata theory on minimal models (see Chapter II). However, the problem is unsolved in general (see §19 in Chapter N), especially when char(s) > 0. 3) Proposition. ladder. 5), we infer that is simply generated for each Remark. g(V, L) 4(V, L) a 4(Vj, L) 2 4(V1, L) 1 0. A(V, L) = 0, the ladder is regular and V1 = 1P1. has a is simply generated. simply generated. Lj if If in addition the equality holds, then Proof. When 4(V, L) 2 0 j.