By Janos Kollár, Shigefumi Mori

One of many significant discoveries of the earlier twenty years in algebraic geometry is the conclusion that the idea of minimum types of surfaces might be generalized to raised dimensional types. This generalization, known as the minimum version software, or Mori's application, has constructed right into a robust instrument with functions to various questions in algebraic geometry and past. This publication presents the 1st complete advent to the circle of rules built round the application, the necessities being just a easy wisdom of algebraic geometry. will probably be of significant curiosity to graduate scholars and researchers operating in algebraic geometry and comparable fields.

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**Sample text**

We can choose a(D) such that (R · (D' + a(D)E)) = 0. Finally choose m such that D" := m(D' + a(D)E) is Cartier. D" has a good chance of being the pull back of a Cartier divisor fJ from Y. If this is indeed the case then mD "" fJ and so D is Q-Cartier. = Case 3. Small contractions. This is a new situation. It could never hap pen for surfaces for dimensional reasons, and it did not happen for smooth threefolds X. We claim that in this case, Y has 'very bad' sin gularities where no multiple of Ky is Cartier.

Ox(kL) is generated by global sections for some k > 0. By Step 1 there is a non-zero s E H0 (X, Ox(mL)) for some m > 0. Rational Curves and the Canonical Class 32 Let D := (s = 0) � X. Then Oy 9:! im[Ox ...!. Ox(mL)) = ker[Ox(mL) -+ OD(mL)) , for some closed subscheme Y of X. We note that Y '# X i s possible since s may vanish on some component of X. We have an exact sequence 0 -+ Oy ((k - l)mL) -+ Ox(kmL) -+ OD(kmL) -+ 0. Since OD (kmL) is very ample for k » 0, it is enough to show that H0(X, Ox(kmL)) ---* H0(D, OD(kmL)) is surjective for k � O.

Let NE(X/Y) C N1 (X/Y) be the cone generated by the 46 Introduction to the Minimal Model Progrom classes of effective curves which are contracted by f. The relative notions NE(X/Y), NE(X/Y)D>o , etc. are defined similarly to the absolute cases. If X and Y are proper, the intersection number provides a pairing N1 (X/Y) x (NS(X)/f*NS(Y)) ®z lR -t JR. In general this pairing is not perfect since the two sides may even have different dimensions (cf. 46)). It is easy to see that N1 (X/Y) is dual to (NS(X)/J* NS(Y)) ®z JR if f is birational and Y is Q-factorial.