Quartic surface

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.

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, MR   1097176
• Jessop, C. M. (1916), Quartic surfaces with singular points, Cornell University Library, ISBN   978-1-4297-0393-2