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 $f$ is a polynomial of degree 4, such as $f(x,y,z) = x^4 + y^4 + xyz + z^2 - 1$. This is a surface in affine space $A3$.

On the other hand, a projective quartic surface is a surface in projective space $P3$ of the same form, but now $f$ is a homogeneous polynomial of 4 variables of degree 4, so for example $f(x,y,z,w) = x^4 + y^4 + xyzw + z^2 w^2 - w^4$.

If the base field is $\mathbb{R}$ or $\mathbb{C}$ 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 $\mathbb{C}$, and quartic surfaces over $\mathbb{R}$. For instance, the Klein quartic is a real surface given as a quartic curve over $\mathbb{C}$. 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