quartic surface

{{Short description|Surface described by a 4th-degree polynomial}}

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

:$f(x,y,z)=0\backslash$

where {{mvar|f}} is a polynomial of degree 4, such as {{tmath|1=f(x,y,z) = x^4 + y^4 + xyz + z^2 - 1}}. This is a surface in affine space {{math|A{{sup|3}}}}.

On the other hand, a projective quartic surface is a surface in projective space {{math|P{{sup|3}}}} of the same form, but now {{mvar|f}} is a homogeneous polynomial of 4 variables of degree 4, so for example {{tmath|1=f(x,y,z,w) = x^4 + y^4 + xyzw + z^2 w^2 - w^4}}.

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