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| A '''cubic surface''' is a [[projective variety]] studied in [[algebraic geometry]]. It is an [[algebraic surface]] in three-dimensional [[projective space]] defined by a single [[quaternary cubic]] [[polynomial]] which is [[homogeneous polynomial|homogeneous]] of degree 3 (hence, cubic). Cubic surfaces are [[del Pezzo surface]]s.
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| [[File:Cubic surface.jpg|thumb|right|Cubic surface]]
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| ==Examples==
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| If <math>\mathbb{P}^3</math> has [[homogeneous co-ordinates]] <math>[X:Y:Z:W]</math>, then the set of points where
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| :<math> X^3 + Y^3 + Z^3 + W^3 = 0</math>
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| is a cubic surface called the [[Fermat cubic surface]]. | |
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| The [[Clebsch surface]] is the set of points where
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| :<math> X^3 + Y^3 + Z^3 + W^3 = (X+Y+Z+W)^3</math>
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| [[Cayley's nodal cubic surface]] is the set of points where
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| :<math>WXY + XYZ + YZW + ZWX =0</math>
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| ==27 lines on a cubic surface==
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| The Cayley-Salmon theorem states that a [[smooth variety|smooth]] cubic surface over an [[algebraically closed field]] contains 27 straight lines. These can be characterized independently of the embedding into projective space as the rational lines with self-intersection number −1, or in other words the −1-curves on the surface. An '''Eckardt point''' is a point where 3 of the 27 lines meet.
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| A smooth cubic surface can also be described as a [[rational surface]] obtained by [[blowing up]] six points in the [[projective plane]] in [[general position]] (in this case, “general position” means no three points are aligned and no six are on a conic section). The 27 lines are the exceptional divisors above the 6 blown up points, the proper transforms of the 15 lines in <math>\mathbb{P}^2</math> which join two of the blown up points, and the proper transforms of the 6 conics in <math>\mathbb{P}^2</math> which contain all but one of the blown up points.
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| [[Clebsch]] gave a model of a cubic surface, called the [[Clebsch diagonal surface]], where all the 27 lines are defined over the field '''Q'''[√5], and in particular are all real.
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| ===Related classifications===
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| The 27 lines can also be identified with some objects arising in representation theory. In particular, these 27 lines can be identified with 27 vectors in the dual of the E6 lattice so their configuration is acted on by the [[Weyl group]] of E6. In particular they form a basis of the 27-dimensional [[fundamental representation]] of the group [[E6 (mathematics)|E<sub>6</sub>]].
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| The 27 lines contain 36 copies of the [[Schläfli double six]] configuration.
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| The 27 lines can be identified with the 27 possible charges of [[M-theory]] on a six-dimensional [[torus]] (6 momenta; 15 [[branes|membrane]]s; 6 [[fivebrane]]s) and the group E<sub>6</sub> then naturally acts as the [[U-duality]] group. This map between [[del Pezzo surface]]s and [[M-theory]] on tori is known as [[mysterious duality]].
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| There are other ways of thinking of these 27 lines. For example, if one projects the cubic from a point which is not on any line (most points of the cubic are like this) then we obtain a double cover of the plane branched along a smooth quartic curve. The 27 lines are mapped to 27 out of the 28 [[bitangents of a quartic|bitangents to this quartic curve]]; the 28th line is the image of the [[exceptional locus]] of the blow-up necessary to resolve the indeterminacy of the projection. These two objects (27 lines on the cubic, 28 bitangents on a quartic), together with the 120 tritangent planes of a canonic sextic curve of genus 4, form a "[[ADE classification#Trinities|trinity]]" in the sense of [[Vladimir Arnold]], specifically a form of [[McKay correspondence]],<ref name="arntrin">{{citation | last = le Bruyn | first = Lieven | title = Arnold’s trinities | url = http://www.neverendingbooks.org/index.php/arnolds-trinities.html | date = 17 June 2008 }}</ref><ref>Arnold 1997, p. 13</ref><ref>{{Harv|McKay|Sebbar|2007|loc=p. 11}}</ref> and can be related to many further objects, including E<sub>7</sub> and E<sub>8</sub>, as discussed at ''[[ADE classification#Trinities|trinities]].''
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| ==Singular cubic surfaces==
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| An example of a singular cubic is [[Cayley's nodal cubic surface]]
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| :<math>WXY + XYZ + YZW + ZWX =0</math>
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| with 4 nodal [[Mathematical singularity|singular points]] at <math>[0:0:0:1]</math> and its permutations. Singular cubic surfaces also contain rational lines, and the number and arrangement of the lines is related to the type of the singularity.
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| The singular cubic surfaces were classified by {{harvtxt|Schlafli|1863}}, and his classification was described by {{harvtxt|Cayley|1869}} and {{harvtxt|Bruce|Wall|1979}}
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| ==References==
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| {{Reflist}}
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| *{{Citation | last1=Bruce | first1=J. W. | last2=Wall | first2=C. T. C. | title=On the classification of cubic surfaces | doi=10.1112/jlms/s2-19.2.245 | mr=533323 | year=1979 | journal=Journal of the London Mathematical Society | issn=0024-6107 | volume=19 | issue=2 | pages=245–256}}
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| *{{Citation |last1=Cayley |first1=Arthur |author1-link=Arthur Cayley | title=A Memoir on Cubic Surfaces | jstor=108997 | publisher=The Royal Society | year=1869 | journal=[[Philosophical Transactions of the Royal Society of London]] | issn=0080-4614 | volume=159 | pages=231–326}}
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| *{{Citation|last=Dolgachev|first=Igor|url=http://www.math.lsa.umich.edu/~idolga/lecturenotes.html|title=Topics in classical algebraic geometry}}
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| *{{Citation | last1=Dolgachev | first1=Igor V. | title=Luigi Cremona (1830–1903) | url=http://www.math.lsa.umich.edu/~idolga/cremona.pdf | publisher=Istituto Lombardo di Scienze e Lettere, Milan | series=Incontro di Studio | mr=2305952 | year=2005 | volume=36 | chapter=Luigi Cremona and cubic surfaces | pages=55–70}}
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| *{{Citation | last1=Henderson | first1=Archibald | title=The twenty-seven lines upon the cubic surface | publisher=Merchant books | series=Reprinting of Cambridge Tracts in Mathematics and Mathematical Physics, No. 13 | isbn=978-1-60386-066-6 | mr=0119139 | year=2007}}
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| *{{Citation | last1=Hunt | first1=Bruce | title=The geometry of some special arithmetic quotients | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-61795-2 | doi=10.1007/BFb0094399 | mr=1438547 | year=1996 | volume=1637}}
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| *{{eom|id=C/c027270|first=V.A.|last= Iskovskikh}}
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| *{{Citation | last1=Manin | first1=Yuri Ivanovich | author1-link=Yuri Ivanovich Manin | title=Cubic forms | publisher=North-Holland | location=Amsterdam | edition=2nd | series=North-Holland Mathematical Library | isbn=978-0-444-87823-6 | mr=833513 | year=1986 | volume=4}}
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| *{{Citation |title=On the Distribution of Surfaces of the Third Order into Species, in Reference to the Absence or Presence of Singular Points, and the Reality of Their Lines | jstor=108795 | publisher=The Royal Society | year=1863 | journal=[[Philosophical Transactions of the Royal Society of London]] | issn=0080-4614 | volume=153 | pages=193–241 | author1=Schläfli, Dr}}
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| *{{Citation | last1=Segre | first1=Beniamino | title=The Non-singular Cubic Surfaces | publisher=[[Oxford University Press]] | mr=0008171 | year=1942}}
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| ==External links==
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| {{commons category|Cubic surfaces}}
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| * {{Citation|first=Oliver|last= Labs|url=http://www.cubics.algebraicsurface.net/ |title=Cubic surfaces home page}}
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| * {{MacTutor Biography|class=HistTopics|id=Cubic_surfaces}}
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| * {{MathWorld |title= Cubic surface|urlname=CubicSurface}}
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| * [http://demonstrations.wolfram.com/LinesOnACubicSurface/ Lines on a Cubic Surface] by Ryan Hoban (The Experimental Geometry Lab at the University of Maryland) based on work by William Goldman, [[The Wolfram Demonstrations Project]].
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| * [http://www.madore.org/cubic-dvd/ The ''Cubic Surfaces'' DVD] (54 animations of cubic surfaces, downloadable separately or as a DVD)
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| {{DEFAULTSORT:Cubic Surface}}
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| [[Category:Algebraic surfaces]]
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| [[Category:Complex surfaces]]
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