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

[math]\displaystyle{ f(x,y,z)=0\ }[/math]

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

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

Special quartic surfaces

• Dupin cyclides
• The Fermat quartic, given by x⁴ + y⁴ + z⁴ + w⁴ =0 (an example of a K3 surface).
• More generally, certain K3 surfaces are examples of quartic surfaces.
• Kummer surface
• Plücker surface
• Weddle surface

See also

• Quadric surface (The union of two quadric surfaces is a special case of a quartic surface)
• Cubic surface (The union of a cubic surface and a plane is another particular type of quartic surface)

References

• Hudson, R. W. H. T. (1990), Kummer's quartic surface, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-39790-2, https://archive.org/details/184605691 
• Jessop, C. M. (1916), Quartic surfaces with singular points, Cornell University Library, ISBN 978-1-4297-0393-2, http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=04290002