Question

Given a vectorspace $V$ (over a field $F$) with a specified basis $b_1, \ldots b_n$ and a set $S \subset V$ with two properties:

1) $S$ is a union of lines through the origin (so for all $s \in S$ we have that $Fs \subset S$)

2) For every affine line line parallel to one of the basis vectors intersects $S$ in exactly one point. (So for every $i \in {1, \ldots n}$ and every $v \in V$ we have $|(v + Fb_i) \cap S| = 1$.)

Is $S$ necessarily a codimension 1 linear subspace or are there other examples?

Answer 1

I think this is a counterexample in $\mathbb{R}^3$:

Start with the plane with normal direction $(1,1,1)$. Tilt each radial ray of that plane by an angle $\alpha = \epsilon \sin 3\phi$ towards $(1,1,1)$ (or away if negative), where $\phi$ is the angle the ray makes with the ray through, say, $(1,-1,0)$. For small $\epsilon$, the resulting surface is still a graph of a single valued function $x=f_1(y,z)$ or $y=f_2(z,x)$ or $z=f_3(x,y)$.

Answer 2

In ${\mathbb R}^3$ with the standard basis try the surface $ x^3 + y^3 + z^3 = 0$.