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?

`$i\in \{1,\ldots,n\}$`

. What about some sort of ruled surface? – David Roberts Jan 4 '12 at 23:48