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 where f is a polynomial of degree 4, such as
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
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
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.