By Takao Fujita

Utilizing options from summary algebraic geometry which were built over contemporary many years, Professor Fujita develops type theories of such pairs utilizing invariants which are polarized higher-dimensional types of the genus of algebraic curves. the center of the e-book is the idea of D-genus and sectional genus constructed via the writer, yet quite a few similar subject matters are mentioned or surveyed. Proofs are given in complete within the imperative a part of the advance, yet history and technical effects are often sketched in while the main points are usually not crucial for figuring out the most important principles.

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30]. ) = 0, 2) is clear by Bertini's theorem. ) > 0, we need to be a little bit more careful. g. 7)] for details. 4) Lemma. a variety D1 U D2 Then Y with Let D be an effective ample divisor on dim Y = n and suppose that for some proper closed subsets dim(DI n D2) a n - 2. Di of Supp(D) _ Supp(D). §4: Existence of a ladder 33 This is clear when Proof. divisor is connected. 1] for n. details. 2), Step 1. We will prove the following more general assertion by induction on 4(V, A) > dim Bs A [A] = L for any linear system A such that is ample.

2) any general member n = 2. is smooth if Indeed, if This is true even if d > 1. 2). Since C C' by Bertini's theorem. C n C' is Hence C Thus C is a smooth elliptic curve d. We claim while the map bijective. Indeed, otherwise, b1(X) 2 2 H1(X, 0) = 0. h: H1(C; Z) - H1(X; Z) Lefschetz theorem. Since Therefore is surjective by the H1(C; Z) = Z ® Z, h Alb(C) = Alb(X). fiber of the Albanese map a: X --* Alb(X). 1P1 for any general F must be F be any Then F n C Let a simple point by the above reasoning and thus F ti d = 1.

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 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.

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