Quartic surface
# 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 $A^3.$

On the other hand, a projective quartic surface is a surface in projective space $P^3$ 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 $x^4$ + $y^4$ + $z^4$ + $w^4$ =0 (an example of a K3 surface).
* More generally, certain K3 surfaces are examples of quartic surfaces.
* Kummer surface
* Plücker surface
* Weddle surface
