Quartic surface

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 Template:Mvar is a polynomial of degree 4, such as Template:Tmath. This is a surface in affine space Template:Math.

On the other hand, a projective quartic surface is a surface in projective space Template:Math of the same form, but now Template:Mvar is a homogeneous polynomial of 4 variables of degree 4, so for example Template:Tmath.

If the base field is Template:Tmath or Template:Tmath 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 Template:Tmath, and quartic surfaces over Template:Tmath. For instance, the Klein quartic is a real surface given as a quartic curve over Template:Tmath. 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

See also

• Quadric surface (The union of two quadric surfaces is a special case of a quartic surface)
• Cubic surface (The union of a cubic surface and a plane is another particular type of quartic surface)

References

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