Geometric-Notes.smooth-cubic-surface.md

= Geometric introduction into the branch curve of a smooth cubic surface =

(Under generic projection)

Consider a three-dimensional projective space $P^3$ with homogeneous coordinates $(x, y, w, z)$.

A cubic surface $S$ in the projective space is given, after a linear change of coordinates, by an equation of the form

$$f(z) = z^3 − 3 a z + b = 0,$$

where a and b are homogeneous forms in $`(x, y, w) $ of degrees 2 and 3, and the projection to the projective pane is is given by

$$(x, y, w, z) → (x, y, w).$$

In these coordinates the ramification curve R is given by the ideal

$$(f, f') = (f, z^2 − a) = (z^3 − 12 b, z^2 − a)$$

and the branch curve $B$, the image of R under the projection, is given by the discriminant

$$∆(f) = b^2 − 4 a^3$$

In particular, one can easily see that the branch curve has six cusps at the intersection of the plane conic defined by $a$ and the plane cubic defined by $b$,

(Besides the usual algebraic interpretation of the condition $(f = f_z = 0)$ on the polynomial f to have a multiple root in z, there is also a nice geometric interpretation of the ramification curve R being the intersection of the surface S and the polar surface S' with respect to the projection center O; the curve R consists of all the points on the surface S where the line joining p and the projection center o is tangent to the surface.)

We give two geometric illustrations (visualisations) where all the coefficients are defined over real numbers (and, moreover, real algebraic numbers).

(These topic is completely classical; this particular illustration was written for the paper "On ramified covers of the projective plane I: Segre's theory and classification in small degrees, with Appendix by Eugenii Shustin", by Michael Friedman, Maxim Leyenson and Eugenii Shustin, on Arxiv.