Birational Geometry of Algebraic Varieties by Janos Kollár, Shigefumi Mori

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.

Show description

Read Online or Download Birational Geometry of Algebraic Varieties PDF

Similar geometry and topology books

Advances in Multiresolution for Geometric Modelling

Multiresolution tools in geometric modelling are thinking about the new release, illustration, and manipulation of geometric items at numerous degrees of aspect. functions comprise quickly visualization and rendering in addition to coding, compression, and electronic transmission of 3D geometric gadgets. This ebook marks the end result of the four-year EU-funded learn venture, Multiresolution in Geometric Modelling (MINGLE).

Spaces of Constant Curvature

This booklet is the 6th variation of the vintage areas of continuous Curvature, first released in 1967, with the former (fifth) version released in 1984. It illustrates the excessive measure of interaction among team conception and geometry. The reader will enjoy the very concise remedies of riemannian and pseudo-riemannian manifolds and their curvatures, of the illustration concept of finite teams, and of symptoms of modern growth in discrete subgroups of Lie teams.

Extra info for Birational Geometry of Algebraic Varieties

Sample text

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'.

Download PDF sample

Birational Geometry of Algebraic Varieties by Janos Kollár, Shigefumi Mori
Rated 4.90 of 5 – based on 8 votes