Torification and Factorization of Birational Maps
نویسندگان
چکیده
Building on work of the fourth author in [68], we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Preliminary version, April 22, 1999.
منابع مشابه
A Note on the Factorization Theorem of Toric Birational Maps after Morelli and Its Toroidal Extension
Building upon the work of [Morelli1,2], we give a coherent presentation of Morelli’s algorithm for the weak and strong factorization of toric birational maps. We also discuss its toroidal extension, which plays a crucial role in the recent solutions [W lodarczyk2,3, Abramovich-Karu-Matsuki-W lodarczyk] of the weak factorization conjecture of general birational maps.
متن کاملRational Singularities, with Applications to Algebraic Surfaces and Unique Factorization
§ o. Some terminology and notation . 196 198 I. Applications to the birational theory of surfaces · . . .. . .. .. . . . . .. . . . 199 § I. Birational behavior of rational singularities . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . 199 § 2. Resolution of singularities by quadratic transformations and normalization (method of Zariski) . . . . . . . . . . . . . . ...
متن کاملErratum to the Paper “a Note on the Factorization Theorem of Toric Birational Maps after Morelli and Its Toroidal Extension”
1. the failure of the algorithm in [AMR] and [Morelli1] for the strong factorization pointed out by Kalle Karu, 2. the statement of a refined weak factorization theorem for toroidal birational morphisms in [AMR], in the form utilized in [AKMR] for the proof of the weak factorization theorem for general birationla maps, avoiding the use of the above mentioned algorithm for the strong factorizati...
متن کاملLocal Strong Factorization of Birational Maps
The strong factorization conjecture states that a proper birational map between smooth algebraic varieties over a field of characteristic zero can be factored as a sequence of smooth blowups followed by a sequence of smooth blowdowns. We prove a local version of the strong factorization conjecture for toric varieties. Combining this result with the monomialization theorem of S. D. Cutkosky, we ...
متن کاملLectures on Factorization of Birational Maps
This is an expanded version of the notes for the lectures given by the author at RIMS in the summer of 1999 to provide a detailed account of the proof for the (weak) factorization theorem of birational maps by Abramovich-Karu-MatsukiW lodarczyk [1]. All the main ideas of these lecture notes, therefore, are derived from the collaboration and joint work with D. Abramovich, K. Karu and J. W lodarc...
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