Enriques surface - Revision history

en>Colonies Chris: sp, date & link fixes; unlinking common words, replaced: , MA → , Massachusetts, Enriques-Kodaira → Enriques–Kodaira using AWB

2014-03-13T19:11:59Z

sp, date & link fixes; unlinking common words, replaced: , MA → , Massachusetts, Enriques-Kodaira → Enriques–Kodaira using AWB

← Older revision		Revision as of 21:11, 13 March 2014
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	~~[http://xrambo.com/blog/192034 at home std testing] Greetings! I am Marvella and I really feel comfortable when individuals use~~ the ~~complete title~~. ~~Hiring is her day [http://www.hotporn123.com/user/KGottscha www.hotporn123.com] occupation now~~ and ~~she will at home std test not alter~~ it ~~anytime quickly~~. His ~~family members lives in South Dakota~~ but ~~his over the [http://samedaystdtesting.com/testing-clinics/florida-fl/hialeah-std-testing/2750-west-68th-street-suite-225-226-33016 counter std] test wife desires them~~ to ~~transfer. The thing she adores most is body building~~ and ~~now she is trying to earn cash with at home std test~~ it.<br><br>~~Have a look at~~ my ~~site ... over the counter std test;~~ [http://www.~~videokeren~~.com/~~blog~~/~~14593 click the up coming website~~],		The writer is known by the name of Numbers Wunder. Years ago we moved to Puerto Rico and my family members enjoys it. I am a meter reader. His wife doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been doing it for fairly a while.<br><br>Also visit my webpage - [http://www.zs-imports.com/blogs/post/70017 www.zs-imports.com]

en>Frietjes: cleanup (wikitables, html markup, layout, etc.)

2014-02-05T23:38:11Z

cleanup (wikitables, html markup, layout, etc.)

← Older revision		Revision as of 01:38, 6 February 2014
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	In [[algebraic geometry]], a '''Fano variety''', introduced in {{harvs\|authorlink=Gino Fano\|last=Fano\|year1=1934\|year2=1942}}, is a [[Complete algebraic variety\|complete variety]] whose [[anticanonical bundle]] is [[ample line bundle\|ample]].		[http://xrambo.com/blog/192034 at home std testing] Greetings! I am Marvella and I really feel comfortable when individuals use the complete title. Hiring is her day [http://www.hotporn123.com/user/KGottscha www.hotporn123.com] occupation now and she will at home std test not alter it anytime quickly. His family members lives in South Dakota but his over the [http://samedaystdtesting.com/testing-clinics/florida-fl/hialeah-std-testing/2750-west-68th-street-suite-225-226-33016 counter std] test wife desires them to transfer. The thing she adores most is body building and now she is trying to earn cash with at home std test it.<br><br>Have a look at my site ... over the counter std test; [http://www.videokeren.com/blog/14593 click the up coming website],

	~~Fano varieties are quite rare, compared to other families, like [[Calabi–Yau manifold]]s and [[general type surface]]s.~~

	~~==The example of projective hypersurfaces==~~

	~~The fundamental example of Fano varieties are the [[algebraic geometry of projective spaces\|projective spaces]]~~: ~~the [[canonical bundle\|anticanonical line bundle]] of <math>\mathbb P_{\mathbf k}^n<~~/~~math> is <math>\mathcal O(n+1)<~~/~~math>, which is [[very ample]] (its [[curvature]] is ''n+1'' times the [[Fubini–Study]] symplectic form)~~.

	~~Let ''D'' be a smooth [[Weil divisor]] in <math>\mathbb P_{\mathbf k}^n<~~/~~math>, from the [[adjunction formula]], we infer <math>\mathcal K_D = (\mathcal K_X + D) = (-(n+1) H + \mathrm{deg}(D) H)_D<~~/~~math>, where ''H'' is the class of the hyperplane. The [[hypersurface~~]~~] ''D'' is therefore Fano if~~ and ~~only if <math>\mathrm{deg}(D) < n+1</math>~~.

	~~==Some properties==~~

	~~The existence of some ample line bundle on ''X''~~ is ~~equivalent to ''X'' being a~~ [~~[projective variety]], so a Fano variety is always projective. The [[Kodaira vanishing theorem]] implies that the [[sheaf cohomology\|higher cohomology groups]] <math>H^i(X, \mathcal O_X)<~~/~~math> of the [[structure sheaf]] vanish for <math>i > 0<~~/~~math>~~. ~~In particular, the [[first Chern class]] induces an isomorphism <math>c_1 : \mathrm{Pic}(X) \to H^2(X,\mathbb Z)~~.</~~math>~~

	~~A Fano variety is simply connected and is [[uniruled]], in particular it has [[Kodaira dimension]] −∞~~.

	~~==Classification in small dimensions==~~

	~~Fano varieties in dimensions 1 are [[isomorphism\|isomorphic]] to the [[projective line]]~~.

	~~In dimension 2 they are the [[del Pezzo surface~~]]s and ~~in the smooth case, they are isomorphic to either <math>\mathbb{P}^1 \times \mathbb{P}^1</math> or to the projective plane blown up in~~ at ~~most 8 general points, and~~ in ~~particular are again all rational.~~

	~~In dimension 3~~ ~~there are non-rational examples. Iskovskih {{harvtxt\|last1=Iskovskih\|year1=1977\|year2=1978\|year3=1979}} classified~~ the ~~smooth Fano 3-folds with second [~~[~~Betti number]] 1 into 17 classes, and {{harvtxt\|Mori\|Mukai\|1981}} classified the smooth ones with second Betti number at least 2, finding 88 deformation classes.~~

	~~==References==~~
	*{{citation\|first=G. \|last=Fano\|chapter=Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulli\|title= Proc. Internat. Congress Mathematicians (Bologna) , 4 , Zanichelli \|year=1934\|pages= 115–119}}
	*{{Citation \| doi=10.1007/BF02565618 \| last1=Fano \| first1=Gino \| title=Su alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche \| url=http://~~gdz~~.~~sub.uni~~-~~goettingen.de~~/~~no_cache~~/~~dms/load/img/?IDDOC=209966 \| mr=0006445 \| year=1942 \| journal=Commentarii Mathematici Helvetici \| issn=0010~~-~~2571 \| volume=14 \| pages=202–211}}~~
	*{{Citation \| last1=Iskovskih \| first1=V. A. \| title=Fano threefolds. I \| doi=10.1070/~~IM1977v011n03ABEH001733 \| mr=463151 \| year=1977 \| journal=Math. USSR~~-~~Izv. \| issn=0373~~-~~2436 \| volume=11 \| issue=3 \| pages=485–527}}~~
	*{{Citation \| last1=Iskovskih \| first1=V. A. \| title=Fano 3-~~folds II \| doi=10.1070/IM1978v012n03ABEH001994Fano+3~~-~~folds+II \| mr=0463151 \| year=1978 \| journal=Math Ussr Izv \| volume=12 \| issue=3 \| pages=469–506}}~~
	*{{Citation \| last1=Iskovskih \| first1=V. A. \| title=Current problems in mathematics, Vol. 12 (Russian) \| publisher=VINITI, Moscow \| mr=537685 \| year=1979 \| chapter=Anticanonical models of three-~~dimensional algebraic varieties \| pages=59–157}}~~
	*{{eom\|id=F/f038220\|first=Vik.S.\|last= Kulikov}}
	*{{Citation \| last1=Mori \| first1=Shigefumi \| last2=Mukai \| first2=Shigeru \| authorlink=Shigefumi Mori \| author2-~~link=Shigeru Mukai \| title=Classification of Fano 3~~-~~folds~~ with B<~~sub~~>2<~~/sub~~>~~≥2 \| doi=10~~.~~1007~~/~~BF01170131 \| mr=641971 \| year=1981 \| journal=Manuscripta Mathematica \| issn=0025-2611 \| volume=36 \| issue=2 \| pages=147–162}}~~
	*{{Citation \| last1=Mori \| first1=Shigefumi \| last2=Mukai \| first2=Shigeru \| authorlink=Shigefumi Mori \| author2-link=Shigeru Mukai \| title=Erratum: "Classification of Fano 3-folds with B<sub>2</~~sub>≥2" \| doi=10~~.~~1007~~/~~s00229-002-0336-2 \| mr=1969009 \| year=2003 \| journal=Manuscripta Mathematica \| issn=0025-2611 \| volume=110 \| issue=3 \| pages=407}}~~

	~~[[Category:Algebraic geometry]]~~
	~~[[Category:Threefolds]~~]

en>R.e.b.: /* Examples */ Expanding article

2013-08-11T02:46:23Z

Examples: Expanding article

← Older revision		Revision as of 04:46, 11 August 2013
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	~~Myrtle Benny~~ is ~~how I~~'~~m known as and I feel comfy when individuals use~~ the ~~full title~~. ~~For many years he~~'~~s been residing~~ in ~~North Dakota~~ and ~~his family loves it~~. ~~Bookkeeping~~ is ~~what I do~~. ~~To gather coins is one~~ of the ~~issues I adore most~~.<br><br>~~Also visit my site~~; [http://~~www~~.~~reachoutsociety~~.~~info~~/~~blog~~/~~435 www~~.~~reachoutsociety~~.~~info~~]		In [[algebraic geometry]], a '''Fano variety''', introduced in {{harvs\|authorlink=Gino Fano\|last=Fano\|year1=1934\|year2=1942}}, is a [[Complete algebraic variety\|complete variety]] whose [[anticanonical bundle]] is [[ample line bundle\|ample]].

			Fano varieties are quite rare, compared to other families, like [[Calabi–Yau manifold]]s and [[general type surface]]s.

			==The example of projective hypersurfaces==

			The fundamental example of Fano varieties are the [[algebraic geometry of projective spaces\|projective spaces]]: the [[canonical bundle\|anticanonical line bundle]] of <math>\mathbb P_{\mathbf k}^n</math> is <math>\mathcal O(n+1)</math>, which is [[very ample]] (its [[curvature]] is ''n+1'' times the [[Fubini–Study]] symplectic form).

			Let ''D'' be a smooth [[Weil divisor]] in <math>\mathbb P_{\mathbf k}^n</math>, from the [[adjunction formula]], we infer <math>\mathcal K_D = (\mathcal K_X + D) = (-(n+1) H + \mathrm{deg}(D) H)_D</math>, where ''H'' is the class of the hyperplane. The [[hypersurface]] ''D'' is therefore Fano if and only if <math>\mathrm{deg}(D) < n+1</math>.

			==Some properties==

			The existence of some ample line bundle on ''X'' is equivalent to ''X'' being a [[projective variety]], so a Fano variety is always projective. The [[Kodaira vanishing theorem]] implies that the [[sheaf cohomology\|higher cohomology groups]] <math>H^i(X, \mathcal O_X)</math> of the [[structure sheaf]] vanish for <math>i > 0</math>. In particular, the [[first Chern class]] induces an isomorphism <math>c_1 : \mathrm{Pic}(X) \to H^2(X,\mathbb Z).</math>

			A Fano variety is simply connected and is [[uniruled]], in particular it has [[Kodaira dimension]] −∞.

			==Classification in small dimensions==

			Fano varieties in dimensions 1 are [[isomorphism\|isomorphic]] to the [[projective line]].

			In dimension 2 they are the [[del Pezzo surface]]s and in the smooth case, they are isomorphic to either <math>\mathbb{P}^1 \times \mathbb{P}^1</math> or to the projective plane blown up in at most 8 general points, and in particular are again all rational.

			In dimension 3 there are non-rational examples. Iskovskih {{harvtxt\|last1=Iskovskih\|year1=1977\|year2=1978\|year3=1979}} classified the smooth Fano 3-folds with second [[Betti number]] 1 into 17 classes, and {{harvtxt\|Mori\|Mukai\|1981}} classified the smooth ones with second Betti number at least 2, finding 88 deformation classes.

			==References==
			*{{citation\|first=G. \|last=Fano\|chapter=Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulli\|title= Proc. Internat. Congress Mathematicians (Bologna) , 4 , Zanichelli \|year=1934\|pages= 115–119}}
			*{{Citation \| doi=10.1007/BF02565618 \| last1=Fano \| first1=Gino \| title=Su alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche \| url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=209966 \| mr=0006445 \| year=1942 \| journal=Commentarii Mathematici Helvetici \| issn=0010-2571 \| volume=14 \| pages=202–211}}
			*{{Citation \| last1=Iskovskih \| first1=V. A. \| title=Fano threefolds. I \| doi=10.1070/IM1977v011n03ABEH001733 \| mr=463151 \| year=1977 \| journal=Math. USSR-Izv. \| issn=0373-2436 \| volume=11 \| issue=3 \| pages=485–527}}
			*{{Citation \| last1=Iskovskih \| first1=V. A. \| title=Fano 3-folds II \| doi=10.1070/IM1978v012n03ABEH001994Fano+3-folds+II \| mr=0463151 \| year=1978 \| journal=Math Ussr Izv \| volume=12 \| issue=3 \| pages=469–506}}
			*{{Citation \| last1=Iskovskih \| first1=V. A. \| title=Current problems in mathematics, Vol. 12 (Russian) \| publisher=VINITI, Moscow \| mr=537685 \| year=1979 \| chapter=Anticanonical models of three-dimensional algebraic varieties \| pages=59–157}}
			*{{eom\|id=F/f038220\|first=Vik.S.\|last= Kulikov}}
			*{{Citation \| last1=Mori \| first1=Shigefumi \| last2=Mukai \| first2=Shigeru \| authorlink=Shigefumi Mori \| author2-link=Shigeru Mukai \| title=Classification of Fano 3-folds with B<sub>2</sub>≥2 \| doi=10.1007/BF01170131 \| mr=641971 \| year=1981 \| journal=Manuscripta Mathematica \| issn=0025-2611 \| volume=36 \| issue=2 \| pages=147–162}}
			*{{Citation \| last1=Mori \| first1=Shigefumi \| last2=Mukai \| first2=Shigeru \| authorlink=Shigefumi Mori \| author2-link=Shigeru Mukai \| title=Erratum: "Classification of Fano 3-folds with B<sub>2</sub>≥2" \| doi=10.1007/s00229-002-0336-2 \| mr=1969009 \| year=2003 \| journal=Manuscripta Mathematica \| issn=0025-2611 \| volume=110 \| issue=3 \| pages=407}}

			[[Category:Algebraic geometry]]
			[[Category:Threefolds]]

en>AnomieBOT: Dating maintenance tags: {{Source needed}}

2012-05-16T05:23:23Z

Dating maintenance tags: {{Source needed}}

New page

Myrtle Benny is how I'm known as and I feel comfy when individuals use the full title. For many years he's been residing in North Dakota and his family loves it. Bookkeeping is what I do. To gather coins is one of the issues I adore most.<br><br>Also visit my site; [http://www.reachoutsociety.info/blog/435 www.reachoutsociety.info]