Math 2.0

Algebraic geometry Surfaces and higher-dimensional varieties Example

Cubic surface

Builds on Del Pezzo surfaceProjective spaceAdjunction formulaTwenty-seven lines on a cubic surface

Example

A cubic surface is the zero set in projective space ℙ3\mathbb{P}^3 of a homogeneous polynomial of degree 33 in four variables. A concrete smooth one is the Fermat cubic over ℂ\C, X={[x:y:z:w]∈ℙ3:x3+y3+z3+w3=0}.X = \set{[x : y : z : w] \in \mathbb{P}^3 : x^3 + y^3 + z^3 + w^3 = 0}. It is smooth: the partial derivatives 3x2,3y2,3z2,3w23x^2, 3y^2, 3z^2, 3w^2 vanish together only at x=y=z=w=0x = y = z = w = 0, which is not a point of ℙ3\mathbb{P}^3. By the adjunction formula, a smooth surface of degree 33 in ℙ3\mathbb{P}^3 has canonical divisor KX=(3−4)H=−HK_X = (3 - 4) H = -H, where HH is a plane section. So −KX=H-K_X = H is ample, and XX is a del Pezzo surface of degree KX2=H2=3K_X^2 = H^2 = 3.

Remarks

The Fermat cubic contains the line {x=−y, z=−w}\set{x = -y, \ z = -w}, since then x3+y3=0x^3 + y^3 = 0 and z3+w3=0z^3 + w^3 = 0. More generally, for cube roots of unity ζ,ζ′\zeta, \zeta' the lines {x=−ζy, z=−ζ′w}\set{x = -\zeta y, \ z = -\zeta' w} lie on XX, which gives 99 lines. The other two ways of pairing the four variables give 99 more each. So there are 2727 lines in all, as the twenty-seven lines theorem predicts. Every smooth cubic surface over an algebraically closed field is the blow-up of ℙ2\mathbb{P}^2 at six points in general position, so it is a rational surface. The Clebsch surface, given by ∑xi=0\sum x_i = 0 and ∑xi3=0\sum x_i^3 = 0 in ℙ4\mathbb{P}^4, is a smooth cubic surface on which all 2727 lines are defined over ℝ\R.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Algebraically closed field, concept, no evidence levelAlgebraically closed …conceptBlowing up, concept, no evidence levelBlowing upconceptCanonical bundle, concept, no evidence levelCanonical bundleconceptCartier divisor, concept, no evidence levelCartier divisorconceptDel Pezzo surface, concept, no evidence levelDel Pezzo surfaceconceptInvertible sheaf, concept, no evidence levelInvertible sheafconceptProjective space, concept, no evidence levelProjective spaceconceptSmooth morphism, concept, no evidence levelSmooth morphismconceptTwenty-seven lines on a cubic surface, statement, no evidence levelTwenty-seven lines on…statementAdjunction formula, statement, no evidence levelAdjunction formulastatementCubic surface, example, no evidence levelCubic surfaceexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraically closed field concept
  2. Blowing up concept
  3. Canonical bundle concept
  4. Cartier divisor concept
  5. Del Pezzo surface concept
  6. Invertible sheaf concept
  7. Projective space concept
  8. Smooth morphism concept
  9. Twenty-seven lines on a cubic surface statement
  10. Adjunction formula statement
  11. Cubic surface example

Further reading

  1. Web https://en.wikipedia.org/wiki/Cubic_surface
About this page
TagKDEVJ69P10
URIm2://KDEVJ69P10
Kindexample
WritingW0 stub
MSC14J26, 14J70
Instance ofDel Pezzo surface
UsesDel Pezzo surface, Projective space, Adjunction formula, Twenty-seven lines on a cubic surface
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