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Quartic surface

Surface described by a 4th-degree polynomial

Summary

Surface described by a 4th-degree polynomial

In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.

More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form :f(x,y,z)=0\ where f is a polynomial of degree 4, such as . This is a surface in affine space A.

On the other hand, a projective quartic surface is a surface in projective space P of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example .

If the base field is or the surface is said to be real or complex respectively. One must be careful to distinguish between algebraic Riemann surfaces, which are in fact quartic curves over , and quartic surfaces over . For instance, the Klein quartic is a real surface given as a quartic curve over . If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.

Special quartic surfaces

  • Dupin cyclides
  • The Fermat quartic, given by x4 + y4 + z4 + w4 =0 (an example of a K3 surface).
  • More generally, certain K3 surfaces are examples of quartic surfaces.
  • Kummer surface
  • Plücker surface
  • Weddle surface

References

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

complex-surfaces algebraic-surfaces
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