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

${\displaystyle f(x,y,z)=0\ }$

where f is a polynomial of degree 4, such as ${\displaystyle f(x,y,z)=x^{4}+y^{4}+xyz+z^{2}-1}$. 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 ${\displaystyle f(x,y,z,w)=x^{4}+y^{4}+xyzw+z^{2}w^{2}-w^{4}}$.

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