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THEORY OF EXCEPTIONAL CURVES
THE PROBLEM OF MINIMAL MODELS
absolutely irreducible affine varieties algebraic closure algebraic extension algebraic surfaces algebraic system algebraically independent antiregular birational transformation antiregular transform arithmetic genus assume belong birational class birationally equivalent biregular biregularly equivalent carrier center Q coefficients completes the proof composite Corollary cycle on F D>jk denote divisorial cycle dominates exceptional variety exists extension of F F-+F field of definition fixed components follows fundamental point fundamental variety Hence hyperplane section incidence correspondence integer intersection involution irreducible components irreducible curve irreducible exceptional curve Lemma Let F linear system locally quadratic transformation maximal contraction maximal ideal non-homogeneous coordinates non-singular surface point of Wjk point Q polynomial prime cycle prime divisor projective model proper transform Proposition prove rational transformation regular at Q relatively minimal model second kind simple point singular points strongly minimal surface F theorem of Bertini tion total transform tr.d transformation of F transformation with center V-+V Zariski zero