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
where f is a polynomial of degree 4, such as f(x,y,z) = x4 + y4 + xyz + z2 − 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) = x4 + y4 + xyzw + z2w2 − w4.
If the base field is R or 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 C, and quartic surfaces over R. For instance, the Klein quartic is a real surface given as a quartic curve over C. If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.