By Janos Kollár, Shigefumi Mori

One of many significant discoveries of the prior 20 years in algebraic geometry is the belief that the speculation of minimum types of surfaces may be generalized to raised dimensional types. This generalization, referred to as the minimum version software, or Mori's software, has built right into a strong device with purposes to diversified questions in algebraic geometry and past. This ebook offers the 1st complete advent to the circle of rules constructed round the application, the necessities being just a uncomplicated wisdom of algebraic geometry. it is going to be of significant curiosity to graduate scholars and researchers operating in algebraic geometry and comparable fields.

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Then, J * M is linearly equivalent to g * 0,2 (b) + E ai Ei, where Ei is the exceptional curve above Pi · f* Mic is linearly equivalent to 0. )D { b) + E aiPi "' 0 on D, which is clearly impossible for general choice of the Pi · However, if the Pi are the points of intersection of a quartic curve Q with D, then the linear system IMI spanned by Q and by the quartics of the form C + (line) is birationally transformed to a free linear system g; 1 IMI and it realizes f : X --+ Y as a morphism into a projective space.

The vanishing is known for i > dim f-1 (y), thus assume (* ) for some i > 1. Let u i , · · · , ur be generators of the maximal ideal m11,y and s : per -. F the homomor phism s(ai , · · · , a ) := L:1 u;a1 defined near 1 - 1 (y). Then we have an r exact sequence: F(11D)er -. F(11D) -t Oxv ® F(11D) -t 0. (im s) (11D) = 0 by the inductive hypothesis, thus we get an exact sequence near y for 11 » 0: , Ri - l f,. F(11D)er -. F(11D) -. (0xv ® F)(11D) = 0. F(11D) = O. Thus Vanishing holds for i - 1, proving ( * ).

23. Let k(X) denote the field of rational functions o n X. The local ring 0E, y C k( X) (that is, the local ring of the generic point of E) is a discrete valuation ring which corresponds to a valuation v(E, Y) of k(X). Such valuations of k(X) are called algebmic valuations. ) Let f' : Y' ---+ X be another birational morphism and E' C Y' an irre ducible divisor such that v(E, Y) = v(E', Y'). This holds iff the rational map Y ---+ X - - + Y' is an isomorphism at the generic points e E E and e ' E E'.