By S. Iitaka

The purpose of this publication is to introduce the reader to the geometric concept of algebraic forms, particularly to the birational geometry of algebraic varieties.This quantity grew out of the author's booklet in eastern released in three volumes by means of Iwanami, Tokyo, in 1977. whereas scripting this English model, the writer has attempted to arrange and rewrite the unique fabric in order that even novices can learn it simply with out bearing on different books, corresponding to textbooks on commutative algebra. The reader is barely anticipated to understand the definition of Noetherin jewelry and the assertion of the Hilbert foundation theorem.

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**Example text**

3. As in (iii) above, when dealing with (pre-)log structures we usually omit the map α from the notation and write simply M for the pair (M, α). 4. If (X, M ) is a log scheme then the units M ∗ ⊂ M are by the ∗ via the map α : M → OX . We deﬁnition of log structure identiﬁed with OX ∗ let λ : OX → M be the resulting inclusion. The monoid law on M deﬁnes an ∗ ∗ on M by translation. The quotient M := M/OX has a natural action of OX monoid structure induced by the monoid structure on M . 5. The notion of log structure makes sense in any ringed topos.

Let B be a scheme and A/B an abelian scheme over B. Fix a ﬁnitely generated free abelian group X with associated torus T . 1) deﬁnes a homomorphism c : X → At as follows. 1 out along the homomorphism x : T → Gm . Let Lx denote the corresponding line bundle on A. The identity element of G induces a trivialization of Lx (0) and hence Lx is a rigidiﬁed line bundle. 3) Lx ⊗ Lx for x, x ∈ X. 4) has a natural algebra structure and there is a canonical isomorphism over A G → SpecA (⊕x∈X Lx ). 5) For any a ∈ A(B) there exists ´etale locally on B an isomorphism t∗a Lx → Lx .

4 in the category of fs monoids is given by the saturation (P ⊕R Q)sat of (P ⊕R Q)int . 5 in the category of ﬁne (resp. 6) (resp. Spec((P ⊕R Q)sat → Spec(Z[(P ⊕R Q)sat ])). 4 Summary of Alexeev’s Results For the convenience of the reader we summarize in this section the main results of Alexeev [3]. At various points in the work that follows we have found it convenient to reduce certain proofs to earlier results of Alexeev instead of proving everything “from scratch”. 1. 6] that a reduced scheme P is called seminormal if for every reduced scheme P and proper bijective morphism f : P → P such that for every p ∈ P mapping to p ∈ P the map k(p) → k(p ) is an isomorphism, the morphism f is an isomorphism.