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 ⁠ f ( x , y , z ) = x 4 + y 4 + x y z + z 2 − 1 {\displaystyle 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 + x y z w + z 2 w 2 − w 4 {\displaystyle f(x,y,z,w)=x^{4}+y^{4}+xyzw+z^{2}w^{2}-w^{4}} ⁠.

If the base field is ⁠ R {\displaystyle \mathbb {R} } ⁠ or ⁠ C {\displaystyle \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 ⁠ C {\displaystyle \mathbb {C} } ⁠, and quartic surfaces over ⁠ R {\displaystyle \mathbb {R} } ⁠. For instance, the Klein quartic is a real surface given as a quartic curve over ⁠ C {\displaystyle \mathbb {C} } ⁠. If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.