Let $k = \bar{k}$ a fixed field. I would like to know if there exist hypersurfaces $X \subset \mathbb{A}\_k^n$ that contain no lines. By line I really mean line, and not just rational curve.


I haven't put any restrictions on $X$, but it's still not clear to me that such things exist. Most likely they form a nonempty open subset of some Hilbert scheme if the degree is large enough.



Hypersurfaces containing no lines
First, consider the projective space $P^n$ instead of affine --- if a hypersurface in affine space contains a line then its closure contains the closure of the line which is a line in the projective space. Consider the Grassmannian $G = Gr(2,n+1)$ parameterizing those lines and let $U$ be the tautological rank 2 subbundle on it. The equation of a hypersurface of degree $d$ gives a section of the vector bundle $S^dU^\*$ on $G$ (and vice versa). Lines on the hypersurface are parameterized by the zero locus of the corresponding section. When 
$$
r(S^dU^\*) = d + 1 > 2(n-1) = \dim G
$$
the zero locus of a general section is empty, since the vector bundle is generated by global sections.


