en>Tom.Reding: General cleanup using AWB

2014-12-02T21:29:42Z

General cleanup using AWB

en>1234qwer1234qwer4: /* See also */

2014-02-27T18:53:46Z

en>AnomieBOT: Dating maintenance tags: {{Clarify}}

2013-12-11T13:05:52Z

Dating maintenance tags: {{Clarify}}

← Older revision		Revision as of 15:05, 11 December 2013
Line 1:		Line 1:
	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Andrew~~ and ~~my spouse doesn~~'~~t like it at all~~. ~~Mississippi~~ is ~~exactly where his house~~ is. ~~Invoicing~~ is ~~my occupation~~. ~~My husband doesn~~'~~t like it the way I do but what I truly like doing~~ is ~~caving but I don~~'t have the ~~time recently~~.<br><br>~~Feel free~~ to ~~visit my webpage~~; ~~psychic readings online~~ ([http://~~cpacs~~.org/~~index~~.~~php?document_srl~~=~~90091&mid~~=~~board_zTGg26 click through~~ the ~~up coming website~~])		In [[algebraic geometry]], '''flips''' and '''flops''' are codimension-2 surgery operations arising in the [[minimal model program]], given by [[blowing up]] along a [[relative canonical model\|relative canonical ring]]. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is conjectured that the same is true in higher dimensions.

			==The minimal model program==
			{{main\|Minimal model program}}
			The minimal model program can be summarised very briefly as follows: given a variety <math>X</math>, we construct a sequence of contractions <math>X = X_1\rightarrow X_2 \rightarrow \cdots \rightarrow X_n </math>, each of which contracts some curves on which the canonical divisor <math>K_{X_i}</math> is negative. Eventually, <math>K_{X_n}</math> should become [[numerically effective\|nef]] (at least in the case of nonnegative [[Kodaira dimension]]), which is the desired result. The major technical problem is that, at some stage, the variety <math>X_i</math> may become 'too singular', in the sense that the canonical divisor <math>K_{X_i}</math> is no longer [[Cartier divisor\|Cartier]], so the intersection number <math>K_{X_i} \cdot C</math> with a curve <math>C</math> is not even defined.

			The (conjectural) solution to this problem is the ''flip''. Given a problematic <math>X_i</math> as above, the flip of <math>X_i</math> is a birational map (in fact an isomorphism in codimension 1) <math>f: X_i \rightarrow X_i^+</math> to a variety whose singularities are 'better' than those of <math>X_i</math>. So we can put <math>X_{i+1} = X_i^+</math>, and continue the process.

			Two major problems concerning flips are to show that they exist and to show that one cannot have an infinite sequence of flips. If both of these problems can be solved then the minimal model program can be carried out.
			The existence of flips for 3-folds was proved by {{harvtxt\|Mori\|1988}}. The existence of log flips, a more general kind of flip, in dimension three and four were proved by {{harvs\|txt\|last=Shokurov\|year1=1993 \|year2=2003}}
			whose work was fundamental to the solution of the existence of log flips and other problems in higher dimension.
			The existence of log flips in higher dimensions has been settled by {{harvs\|first=Caucher \|last=Birkar\|first2= Paolo \|last2=Cascini\|first3= Christopher D. \|last3=Hacon\|first4= James\|last4= McKernan\|year=2010}}. On the other hand, the problem of termination—proving that there can be no infinite sequence of flips—is still open in dimensions greater than 3.

			==Definition==
			If ''f'':''X''→''Y'' is a morphism, and ''K'' is the canonical bundle of ''X'', then the relative canonical ring of ''f'' is
			:<math>\oplus_m f_*(\mathcal O_X(mK))</math>
			and is a sheaf of graded algebras over the sheaf ''O''<sub>''Y''</sub> of regular functions on ''Y''.
			The blowup ''f''<sup>+</sup>
			:<math>f^+:X^+= Proj(\oplus_m f_*(\mathcal O_X(mK)))\to Y</math>
			of ''Y'' along the relative canonical ring is a morphism to ''Y''. If the relative canonical ring is finitely generated (as an algebra over ''O''<sub>''Y''</sub>) then the morphism ''f''<sup>+</sup> is called the '''flip''' of ''f'' if −''K'' is relatively ample, and the '''flop''' of ''f'' if ''K'' is relatively trivial. (Sometimes the induced birational morphism from ''X'' to ''X''<sup>+</sup> is called a flip or flop.)

			In applications, ''f'' is often a [[small contraction]] of an extremal ray, which implies several extra properties:
			*The exceptional sets of both maps ''f'' and ''f''<sup>+</sup> have codimension at least 2,
			*''X'' and ''X''<sup>+</sup> only have mild singularities, such as [[terminal singularities]].
			*''f'' and ''f''<sup>+</sup> are birational morphisms onto ''Y'', which is normal and projective.
			*All curves in the fibers of ''f'' and ''f''<sup>+</sup> are numerically proportional.

			==Examples==
			The first example of a flop, known as the '''Atiyah flop''', was found in {{harv\|Atiyah\|1958}}.
			Let ''Y'' be the zeros of ''xy'' = ''zw'' in '''A'''<sup>4</sup>, and let ''V'' be the blowup of ''Y'' at the origin.
			The exceptional locus of this blowup is isomorphic to '''P'''<sup>1</sup>×'''P'''<sup>1</sup>, and can be blown down to '''P'''<sup>1</sup> in 2 different ways, giving varieties ''X''<sub>1</sub> and ''X''<sub>2</sub>. The natural birational map from ''X''<sub>1</sub> to ''X''<sub>2</sub> is the Atiyah flop.

			{{harvtxt\|Reid\|1983}} introduced '''Reid's pagoda''', a generalization of Atiyah's flop replacing ''Y'' by the zeros of
			''xy'' = (''z''+''w''<sup>''k''</sup>)(''z''−''w''<sup>''k''</sup>).

			==References==
			*{{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| title=On analytic surfaces with double points \| mr=0095974 \| year=1958 \| journal=Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences \| volume=247 \| pages=237–244 \| doi=10.1098/rspa.1958.0181 \| issue=1249}}
			*{{Citation \| last1=Birkar \| first1=Caucher \| last2=Cascini \| first2=Paolo \| last3=Hacon \| first3=Christopher D. \| last4=McKernan \| first4=James \| title=Existence of minimal models for varieties of log general type \| arxiv=math.AG/0610203 \| doi=10.1090/S0894-0347-09-00649-3 \| mr=2601039 \| year=2010 \| journal=[[Journal of the American Mathematical Society]] \| issn=0894-0347 \| volume=23 \| issue=2 \| pages=405–468}}
			* {{ citation
			\| last = Corti
			\| first = Alessio
			\| authorlink = Alessio Corti
			\| title = What Is...a Flip?
			\| journal = Notices of the American Mathematical Society
			\| year = 2004
			\| month = December
			\| volume = 51
			\| issue = 11
			\| pages = 1350–1351
			\| url = http://www.ams.org/notices/200411/what-is.pdf
			\| format = [[PDF]]
			\| accessdate = 2008-01-17 }}
			*{{Citation \| authorlink=János_Kollár \| last1=Kollár \| first1=János \| title=Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) \| publisher=Math. Soc. Japan \| location=Tokyo \| mr=1159257 \| year=1991 \| chapter=Flip and flop \| pages=709–714}}
			* {{Citation \| authorlink=János_Kollár \| last1=Kollár \| first1=János \| title=Surveys in differential geometry (Cambridge, MA, 1990) \| publisher=Lehigh Univ. \| location=Bethlehem, PA \| mr=1144527 \| year=1991 \| chapter=Flips, flops, minimal models, etc \| pages=113–199}}
			* {{Citation \| authorlink1=János_Kollár \| last1=Kollár \| first1=János \| authorlink2=Shigefumi Mori \| last2=Mori \| first2=Shigefumi \| title=Birational Geometry of Algebraic Varieties \| publisher=[[Cambridge University Press]] \| year= 1998 \| isbn=0-521-63277-3}}
			*{{Citation \| last1=Matsuki \| first1=Kenji \| title=Introduction to the Mori program \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Universitext \| isbn=978-0-387-98465-0 \| mr=1875410 \| year=2002}}
			*{{Citation \| authorlink=Shigefumi Mori \| last1=Mori \| first1=Shigefumi \| title=Flip theorem and the existence of minimal models for 3-folds \| jstor=1990969 \| mr=924704 \| year=1988 \| journal=[[Journal of the American Mathematical Society]] \| volume=1 \| issue=1 \| pages=117–253}}
			*{{citation\|first=David\|last= Morrison \|url=http://www.math.ucsb.edu/~drm/lectures/ueno-talk.pdf\|title= Flops, flips, and matrix factorization\|publisher=Algebraic Geometry and Beyond, RIMS, Kyoto University \|year=2005}}
			*{{Citation \| last1=Reid \| first1=Miles \| author1-link=Miles Reid \| title=Algebraic varieties and analytic varieties (Tokyo, 1981) \| publisher=North-Holland \| location=Amsterdam \| series=Adv. Stud. Pure Math. \| mr=715649 \| year=1983 \| volume=1 \| chapter=Minimal models of canonical $3$-folds \| pages=131–180}}
			*{{Citation \| last1=Shokurov \| first1=V.V. \| author1-link=Vyacheslav Shokurov \| title=Three-dimensional log flips. With an appendix in English by Yujiro Kawamata
			\| publisher=Russian Acad. Sci. Izv. Math. 40 \| year=1993 \| volume=1 \| pages=95–202.}}
			*{{Citation \| last1=Shokurov \| first1=V.V. \| author1-link=Vyacheslav Shokurov \| title=Prelimiting flips\| publisher=Proc. Steklov Inst. Math. 240 \| year=2003 \| pages=75–213.}}

			[[Category:Algebraic geometry]]
			[[Category:Birational geometry]]

en>Headbomb: Disambiguated: large scale structure → Large-scale structure of the universe (2), CDM → Cold dark matter

2012-08-09T19:47:32Z

Disambiguated: large scale structure → Large-scale structure of the universe (2), CDM → Cold dark matter

New page

I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Mississippi is exactly where his house is. Invoicing is my occupation. My husband doesn't like it the way I do but what I truly like doing is caving but I don't have the time recently.<br><br>Feel free to visit my webpage; psychic readings online ([http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 click through the up coming website])

CMB cold spot - Revision history

en>Tom.Reding: General cleanup using AWB

en>1234qwer1234qwer4: /* See also */

en>AnomieBOT: Dating maintenance tags: {{Clarify}}

en>Headbomb: Disambiguated: large scale structure → Large-scale structure of the universe (2), CDM → Cold dark matter