Cubic surface

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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 of degree 3 (hence, cubic). Cubic surfaces are del Pezzo surfaces.

Cubic surface

Examples

If {\mathbb {P}}^{3} has homogeneous co-ordinates [X:Y:Z:W], then the set of points where

X^{3}+Y^{3}+Z^{3}+W^{3}=0

is a cubic surface called the Fermat cubic surface.

The Clebsch surface is the set of points where

X^{3}+Y^{3}+Z^{3}+W^{3}=(X+Y+Z+W)^{3}

Cayley's nodal cubic surface is the set of points where

WXY+XYZ+YZW+ZWX=0

27 lines on a cubic surface

The Cayley-Salmon theorem states that a 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.

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 {\mathbb {P}}^{2} which join two of the blown up points, and the proper transforms of the 6 conics in {\mathbb {P}}^{2} which contain all but one of the blown up points.

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.

Related classifications

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.

The 27 lines contain 36 copies of the Schläfli double six configuration.

The 27 lines can be identified with the 27 possible charges of M-theory on a six-dimensional torus (6 momenta; 15 membranes; 6 fivebranes) and the group E6 then naturally acts as the U-duality group. This map between del Pezzo surfaces and M-theory on tori is known as mysterious duality.

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 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 "trinity" in the sense of Vladimir Arnold, specifically a form of McKay correspondence,[1][2][3] and can be related to many further objects, including E7 and E8, as discussed at trinities.

Singular cubic surfaces

An example of a singular cubic is Cayley's nodal cubic surface

WXY+XYZ+YZW+ZWX=0

with 4 nodal singular points at [0:0:0:1] 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.

The singular cubic surfaces were classified by Schlafli (1863), and his classification was described by Cayley (1869) and Bruce & Wall (1979)

References

  1. ↑ le Bruyn, Lieven (17 June 2008), Arnold’s trinities 
  2. ↑ Arnold 1997, p. 13
  3. ↑ (McKay & Sebbar 2007, p. 11)
  • Bruce, J. W.; Wall, C. T. C. (1979), "On the classification of cubic surfaces", Journal of the London Mathematical Society 19 (2): 245–256, doi:10.1112/jlms/s2-19.2.245, ISSN 0024-6107, MR 533323 
  • Cayley, Arthur (1869), "A Memoir on Cubic Surfaces", Philosophical Transactions of the Royal Society of London (The Royal Society) 159: 231–326, ISSN 0080-4614, JSTOR 108997 
  • Dolgachev, Igor, Topics in classical algebraic geometry 
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  • Henderson, Archibald (2007), The twenty-seven lines upon the cubic surface, Reprinting of Cambridge Tracts in Mathematics and Mathematical Physics, No. 13, Merchant books, ISBN 978-1-60386-066-6, MR 0119139 
  • Hunt, Bruce (1996), The geometry of some special arithmetic quotients, Lecture Notes in Mathematics 1637, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094399, ISBN 978-3-540-61795-2, MR 1438547 
  • Iskovskikh, V.A. (2001), "C/c027270", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  • Manin, Yuri Ivanovich (1986), Cubic forms, North-Holland Mathematical Library 4 (2nd ed.), Amsterdam: North-Holland, ISBN 978-0-444-87823-6, MR 833513 
  • Schläfli, Dr (1863), "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", Philosophical Transactions of the Royal Society of London (The Royal Society) 153: 193–241, ISSN 0080-4614, JSTOR 108795 
  • Segre, Beniamino (1942), The Non-singular Cubic Surfaces, Oxford University Press, MR 0008171 

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