Cubic surface

In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space ${\displaystyle \mathbf {P} ^{3}}$. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface

${\displaystyle x^{3}+y^{3}+z^{3}+w^{3}=0}$

in ${\displaystyle \mathbf {P} ^{3}}$. Many properties of cubic surfaces hold more generally for del Pezzo surfaces.