27 lines on a cubic surface

A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single quaternary cubic polynomial which is homogeneous of degree 3 (hence, cubic). Cubic surfaces are del Pezzo surfaces.

If ${\displaystyle \mathbb {P} ^{3}}$ has homogeneous co-ordinates ${\displaystyle [X:Y:Z:W]}$, then the set of points where

is a cubic surface called the Fermat cubic surface.

The Clebsch surface is the set of points where

Cayley's nodal cubic surface is the set of points where

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